Startseite Mathematik Some results on the total proper k-connection number
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Some results on the total proper k-connection number

  • Yingbin Ma EMAIL logo und Hui Zhang
Veröffentlicht/Copyright: 8. April 2022
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 20 Heft 1

Abstract

In this paper, we first investigate the total proper connection number of a graph G according to some constraints of G ¯ . Next, we investigate the total proper connection numbers of graph G with large clique number ω ( G ) = n − s for 1 ≤ s ≤ 3 . Finally, we determine the total proper k -connection numbers of circular ladders, Möbius ladders and all small cubic graphs of order 8 or less.

Keywords: total coloring; total proper path; total proper k-connected; total proper k-connection number; complement graph; clique number
MSC 2010: 05C15; 05C35; 05C40

1 Introduction

In this paper, all graphs under our consideration are simple, finite and undirected. We follow the notation and terminology of [1]. For a graph G , we denote by V ( G ) , E ( G ) and diam ( G ) the vertex set, edge set and diameter of G , respectively. The distance between two vertices u and v in a connected graph G , denoted by dist ( u , v ) , is the length of a shortest path between them in G . The eccentricity of a vertex v in G is defined as ecc G ( v ) = max x ∈ V ( G ) dist ( v , x ) . For convenience, a set of internally pairwise vertex disjoint paths will be called disjoint.

In recent years, colored notions of connectivity in graphs becomes a new and active subject in graph theory. Stating from rainbow connection [2], rainbow vertex connection [3] and total rainbow connection [4,5] appeared later. Many researchers are working in this field, and a lot of papers have been published in journals, see [6,7,8, 9,10,11, 12,13,14, 15,16] for details. The reader can also see [17] for a survey, [18] for a dynamic survey and [19] for a new monograph on this topic.

In 2012, Borozan et al. [20] introduced the concept of proper k -connection number. A path in an edge-colored graph is a proper path if any two adjacent edges on the path differ in color. An edge-colored graph is proper k-connected if any two distinct vertices of the graph are connected by k disjoint proper paths. The proper k-connection number of a k -connected graph G , denoted by p c k ( G ) , is defined as the smallest number of colors that are needed in order to make G proper k -connected. For more results, the reader can see [21,22, 23,24] for details.

As a natural generalization, Jiang et al. [25] presented the concept of proper vertex k -connection number. A path in a vertex-colored graph is a vertex proper path if any two internal adjacent vertices of the path differ in color. A vertex-colored graph is proper vertex k-connected if any two distinct vertices of the graph are connected by k disjoint vertex proper paths. For a k -connected graph G , the proper vertex k-connection number of G , denoted by p v c k ( G ) , is defined as the smallest number of colors required to make G proper vertex k -connected.

Motivated by the concept of total chromatic number of graph, now for proper connection and proper vertex connection, the concept of total proper connection was introduced by Jiang et al. [26]. A total coloring of a graph G is a mapping from the set V ( G ) ∪ E ( G ) to some finite set of colors. A path in a total-colored graph is a total proper path if the coloring of the edges and internal vertices is proper, that is, any two adjacent or incident elements of edges and internal vertices on the path differ in color. A total-colored graph is total proper k-connected if any two distinct vertices of the graph are connected by k disjoint total proper paths. For a connected graph G , the total proper k-connection number of a k -connected graph G , denoted by tpc k ( G ) , is defined as the smallest number of colors that are needed in order to make G total proper k -connected. For convenience, we write tpc ( G ) for tpc 1 ( G ) . Obviously, tpc ( G ) ≤ tpc 2 ( G ) ≤ tpc 3 ( G ) . By [26], if G is complete, then tpc ( G ) = 1 ; if G has a Hamiltonian path that is not complete, then tpc ( G ) = 3 . Note that if G is a nontrivial connected graph and H is a connected spanning subgraph of G , then tpc ( G ) ≤ tpc ( H ) .

In this paper, we investigate the total proper connection number of a graph G under some constraints on its complement graph G ¯ .

Theorem 1.1

Let G be a connected graph of order n ≥ 3 , if diam ( G ¯ ) does not belong to the following two cases: (i) diam ( G ¯ ) = 2 , 3 , (ii) G ¯ contain exactly two components and one of them is trivial, then tpc ( G ) ≤ 4 .

For the remaining cases, tpc ( G ) can be very large as discussed in Section 2. Then we add a constraint, i.e., we let G ¯ be triangle-free. Hence, G is claw-free, and we can derive our next main result:

Theorem 1.2

For a connected graph G , if G ¯ is triangle-free, then tpc ( G ) = 3 .

Recall that a clique of a graph is a set of mutually adjacent vertices, and that a maximum clique is a clique of the largest possible size in a given graph. The clique number ω ( G ) of a graph G is the number of vertices in a maximum clique in G . Let G be a connected graph, and let X be a maximum clique of G . We say that N X ( u ) is the set of neighbors of u in G [ X ] and d X ( u ) = ∣ N X ( u ) ∣ . Let F = G [ V ( G ) \ X ] . Kemnitz and Schiermeyer [9] considered graphs with r c ( G ) = 2 and large clique number. In this paper, we characterize graphs with small total proper connection number with respect to their large clique number. If ω ( G ) = n , then G is a complete graph, which implies tpc ( G ) = 1 . If G is connected and ω ( G ) = n − 1 , then G has a Hamiltonian path, and so tpc ( G ) = 3 . For the cases ω ( G ) = n − 2 , n − 3 , we obtain the following three main results.

Figure 1 All connected cubic graphs of order 8.
Figure 1

All connected cubic graphs of order 8.

Theorem 1.3

Let G be a connected graph of order n . If ω ( G ) = n − 2 and X is a maximum clique of G with V ( G ) \ X = { u 1 , u 2 } , then tpc ( G ) = 3 .

Theorem 1.4

Let G be a connected graph of order n , diam ( G ) = 2 . If ω ( G ) = n − 3 and X is a maximum clique of G with V ( G ) \ X = { u 1 , u 2 , u 3 } , then tpc ( G ) = 3 or tpc ( G ) = 4 for the following case F ≅ 3 K 3 , ∣ N X ( u 1 ) ∩ N X ( u 2 ) ∩ N X ( u 3 ) ∣ = 1 and d X ( u 1 ) = d X ( u 2 ) = d X ( u 3 ) = 1 .

Theorem 1.5

Let G be a connected graph of order n , diam ( G ) ≥ 3 . If ω ( G ) = n − 3 and X is a maximum clique of G with V ( G ) \ X = { u 1 , u 2 , u 3 } , then tpc ( G ) = 3 , or tpc ( G ) = 4 and one of the following holds.

  1. F ≅ 3 K 3 , N X ( u ) ∩ N X ( v ) ≠ ∅ and d X ( u ) = d X ( v ) = 1 , where u and v are any two distinct vertices in V ( G ) \ X .

  2. F ≅ 3 K 3 , d X ( u 1 ) = d X ( u 2 ) = d X ( u 3 ) = 1 and for any two vertices in V ( G ) \ X , there is no common neighbor in G [ X ] .

For an integer n ≥ 3 , the circular ladder CL 2 n of order 2 n is a cubic graph constructed by taking two copies of the cycle C n on disjoint vertex sets ( u 1 , u 2 , … , u n ) and ( v 1 , v 2 , … , v n ) , then joining the corresponding vertices u i v i for 1 ≤ i ≤ n . The Möbius ladder M 2 n of order 2 n is obtained from the ladder by deleting the edges u 1 u n and v 1 v n , and then inserting two edges u 1 v n and u n v 1 . Subscripts are considered modulo n , and we can derive our next main result:

Theorem 1.6

Let n be an integer with n ≥ 3 . Then

  1. tpc ( CL 2 n ) = tpc 2 ( CL 2 n ) = 3 , tpc 3 ( CL 2 n ) = 4 .

  2. tpc ( M 2 n ) = tpc 2 ( M 2 n ) = 3 , tpc 3 ( M 2 n ) = 4 .

In [7], Fujie-Okamoto et al. investigated the rainbow k -connection numbers of all small cubic graphs of order 8 or less. In this paper, we determine the total proper k -connection numbers of all small cubic graphs of order 8 or less. We can easily verify that all such cubic graphs have orders 4, 6, or 8, and those with orders 4 or 6 are K 4 , K 3 , 3 , and K 3 □ K 2 (where □ denotes Cartesian product). In [27], it was shown that all connected cubic graphs of order 8 are Q 3 , M 8 , F 1 , F 2 , and F 3 , and these graphs are depicted in Figure 1. Our last main result is stated as follows:

Theorem 1.7

  1. tpc ( K 4 ) = 1 , tpc 2 ( K 4 ) = 3 , tpc 3 ( K 4 ) = 4 .

  2. tpc ( K 3 , 3 ) = tpc 2 ( K 3 , 3 ) = 3 , tpc 3 ( K 3 , 3 ) = 4 .

  3. tpc ( K 3 □ K 2 ) = tpc 2 ( K 3 □ K 2 ) = 3 , tpc 3 ( K 3 □ K 2 ) = 4 .

  4. tpc ( Q 3 ) = tpc 2 ( Q 3 ) = 3 , tpc 3 ( Q 3 ) = 4 .

  5. tpc ( M 8 ) = tpc 2 ( M 8 ) = 3 , tpc 3 ( M 8 ) = 4 .

  6. tpc ( F 1 ) = tpc 2 ( F 1 ) = 3 , tpc 3 ( F 1 ) = 4 .

  7. tpc ( F 2 ) = tpc 2 ( F 2 ) = 3 .

  8. tpc ( F 3 ) = tpc 2 ( F 3 ) = 3 , tpc 3 ( F 3 ) = 4 .

2 Proof of Theorem 1.1

In order to prove Theorem 1.1, we need the following lemmas.

Lemma 2.1

[26] For 2 ≤ m ≤ n , we have tpc ( K m , n ) = 3 .

Lemma 2.2

[26] If G is a complete multipartite graph that is neither a complete graph nor a tree, then tpc ( G ) = 3 .

Let N G ¯ i ( x ) = { v : dist G ¯ ( x , v ) = i } , where 0 ≤ i ≤ 3 , and N G ¯ 4 ( x ) = { v : dist ( x , v ) ≥ 4 } . In this paper, we use N G ¯ i instead of N G ¯ i ( x ) for convenience. Then N G ¯ 0 = { x } and N G ¯ 1 = N G ¯ ( x ) .

Lemma 2.3

For a connected graph G , if G ¯ is connected and diam ( G ¯ ) ≥ 4 , then tpc ( G ) ≤ 4 .

Proof

Choose a vertex x with ecc G ¯ ( x ) = diam ( G ¯ ) . By the definition of N G ¯ i , we know u v ∈ E ( G ) for any u ∈ N G ¯ i , v ∈ N G ¯ j with ∣ i − j ∣ ≥ 2 . Now we define a total coloring of G as follows: assign color 1 to the edges x u for u ∈ N G ¯ 2 , all edges between N G ¯ 1 and N G ¯ 3 , and all vertices and edges in N G ¯ 4 ; assign color 2 to the edges x u for u ∈ N G ¯ 3 , all edges between N G ¯ 2 and N G ¯ 4 , and all vertices and edges in N G ¯ 1 ; assign color 3 to the edges x u for u ∈ N G ¯ 4 , all edges between N G ¯ 1 and N G ¯ 4 , and all vertices and edges in N G ¯ 2 , N G ¯ 3 ; assign color 4 to the vertex x .

We prove that there is a total proper path between any two vertices u and v of G . It is trivial when u v ∈ E ( G ) . Thus, we only need to consider the pairs u , v ∈ N G ¯ i or u ∈ N G ¯ i , v ∈ N G ¯ i + 1 . Note that P = x x 3 x 1 x 4 x 2 is a total proper path, where x i ∈ N G ¯ i . By means of the path P , we can find that u and v are connected by some total proper path for any u ∈ N G ¯ i , v ∈ N G ¯ i + 1 . If i = 1 , then P = u x 3 x x 2 x 4 v is a total proper path, where x i ∈ N G ¯ i . If i = 2 , then P = u x x 4 v is a total proper path, where x 4 ∈ N G ¯ 4 . If i = 3 , then P = u x 1 x 4 x 2 x v is a total proper path, where x i ∈ N G ¯ i . If i = 4 , then P = u x x 2 v is a total proper path, where x 2 ∈ N G ¯ 2 . Hence, tpc ( G ) ≤ 4 .□

Proof of Theorem 1.1

Assume that G ¯ is connected. Since diam ( G ¯ ) ≥ 4 , we have tpc ( G ) ≤ 4 by Lemma 2.3. Assume that G ¯ is disconnected. By the assumption, we know that there exist either at least three connected components or exactly two nontrivial components. Let G i ¯ be the components of G ¯ with t i = ∣ V ( G i ¯ ) ∣ , where 1 ≤ i ≤ h . Then G contains a connected spanning subgraph K t 1 , t 2 , … , t h , and we have tpc ( G ) ≤ tpc ( K t 1 , t 2 , … , t h ) = 3 by Lemma 2.2. Note that G is not complete. Thus, tpc ( G ) ≥ 3 , and so tpc ( G ) = 3 .□

Next, we will give three examples to show that tpc ( G ) can be arbitrarily large if one of the following three conditions holds: diam ( G ¯ ) = 2 , diam ( G ¯ ) = 3 , G ¯ contains exactly two connected components and one of them is trivial.

Figure 2 The graph of Example 2.4.
Figure 2

The graph of Example 2.4.

Example 2.4

For the graph G ¯ shown in Figure 2, we choose a vertex x with ecc G ¯ ( x ) = diam ( G ¯ ) . Let N G ¯ 1 ( x ) = { u i ∣ 1 ≤ i ≤ k } , N G ¯ 2 ( x ) = { v j ∣ 1 ≤ j ≤ k } , and let E ( G ¯ ) = { x u i ∣ 1 ≤ i ≤ k } ∪ { u i 1 u i 2 ∣ 1 ≤ i 1 , i 2 ≤ k } ∪ { v j 1 v j 2 ∣ 1 ≤ j 1 , j 2 ≤ k } ∪ { u i v j ∣ 1 ≤ i , j ≤ k } \ { u i v i ∣ 1 ≤ i ≤ k } , where k ≥ 3 . Obviously, diam ( G ¯ ) = 2 and G is a tree. Then tpc ( G ) = Δ ( G ) + 1 = k + 1 by [26, Theorem 1]. Observe that tpc ( G ) will be arbitrarily large based on the increase of k .

Figure 3 The graph of Example 2.5.
Figure 3

The graph of Example 2.5.

Example 2.5

For the graph G ¯ shown in Figure 3, we choose a vertex x with ecc G ¯ ( x ) = diam ( G ¯ ) . Let N G ¯ 1 ( x ) = { u i ∣ 1 ≤ i ≤ k } , N G ¯ 2 ( x ) = { v j ∣ 1 ≤ j ≤ k } , and N G ¯ 3 ( x ) = { w s ∣ 1 ≤ s ≤ k } , where k ≥ 3 . Furthermore, let E ( G ¯ ) = { x u i ∣ 1 ≤ i ≤ k } ∪ { u i v j ∣ 1 ≤ i , j ≤ k } ∪ { v j w s ∣ 1 ≤ j , s ≤ k } ∪ { v j 1 v j 2 ∣ 1 ≤ j 1 , j 2 ≤ k } . Obviously, diam ( G ¯ ) = 3 and G is a connected graph. Note that N G ¯ 2 ( x ) is a stable set in G ¯ , and each edge between x and N G ¯ 2 ( x ) is a cut edge in G . Therefore, tpc ( G ) ≥ k + 1 by [26, Proposition 2], and so tpc ( G ) will be arbitrarily large based on the increase of k .

Example 2.6

Let G ¯ contains exactly two components G 1 ¯ and G 2 ¯ , where G 1 ¯ is trivial and G 2 ¯ is a clique of G ¯ . Clearly, G is a star, and tpc ( G ) = ∣ V ( G 2 ¯ ) ∣ + 1 . Thus, tpc ( G ) can be made arbitrarily large by increasing ∣ V ( G 2 ¯ ) ∣ .

3 Proof of Theorem 1.2

Lemma 3.1

For a connected graph G , if G ¯ is connected with diam ( G ¯ ) ≥ 4 , and G ¯ is triangle-free, then tpc ( G ) = 3 .

Proof

Choose a vertex x with ecc G ¯ ( x ) = diam ( G ¯ ) . Since G ¯ is triangle-free, we know that N G ¯ 1 is a clique in G . Now we define a total coloring of G as follows: assign color 1 to the edges x u for u ∈ N G ¯ 2 , all edges between N G ¯ 1 and N G ¯ 3 , and all vertices and edges in N G ¯ 4 ; assign color 2 to the edges between N G ¯ 2 and N G ¯ 4 , all vertices and edges in N G ¯ 1 , and the vertex x ; assign color 3 to the edges x u for u ∈ N G ¯ 3 , N G ¯ 4 , all edges between N G ¯ 1 and N G ¯ 4 , and all vertices and edges in N G ¯ 2 , N G ¯ 3 .

We prove that there is a total proper path between any two distinct vertices u and v in G . Note that P = x x 2 x 4 x 1 x 3 is a total proper path, where x i ∈ N G ¯ i . By means of the path P , we can find that u and v are connected by some total proper path for any u ∈ N G ¯ i , v ∈ N G ¯ i + 1 . Thus, we only need to consider the pairs u , v ∈ N G ¯ i . For i = 2 , P = u x x 4 v is a total proper path, where x 4 ∈ N G ¯ 4 . For i = 4 , P = u x x 2 v is a total proper path, where x 2 ∈ N G ¯ 2 . For i = 3 , P = u x 1 x 4 x 2 x v is a total proper path, where x i ∈ N G ¯ i . Thus, G is total proper connected with the above coloring, and so tpc ( G ) = 3 .□

Lemma 3.2

Let G be a connected graph. If diam ( G ¯ ) = 3 and G ¯ is triangle-free, then tpc ( G ) = 3 .

Proof

For a vertex x of G ¯ satisfying ecc G ¯ ( x ) = diam ( G ¯ ) = 3 , let n i represent the number of vertices with distance i from x . If n 1 = n 2 = n 3 = 1 , then G ≅ P 4 , and so tpc ( G ) = 3 .

Case 1. Two of n 1 , n 2 , n 3 are equal to 1. Without loss of generality, we may assume n 1 = n 2 = 1 . Since G ¯ is triangle-free, we have that N G ¯ 3 is a stable set in G ¯ , and so a clique in G . We can find that G has a Hamiltonian path. Thus, tpc ( G ) = 3 .

Case 2. One of n 1 , n 2 , n 3 is equal to 1. Suppose n 2 = 1 . Since G ¯ is triangle-free, we know that N G ¯ 1 and N G ¯ 3 is a stable set in G ¯ , and so a clique in G . Note that G has a Hamiltonian path, and so tpc ( G ) = 3 .

Subcase 2.1. n 1 = 1 . Since G ¯ is triangle-free, we obtain that N G ¯ 2 is a clique in G . Define a total coloring of G as follows: assign color 3 to the vertex x , all edges between N G ¯ 2 and N G ¯ 3 , and all edges between N G ¯ 1 and N G ¯ 3 ; assign color 2 to the edges x u for u ∈ N G ¯ 2 , and all vertices and edges in N G ¯ 3 ; assign color 1 to the edges x u for u ∈ N G ¯ 3 , and all vertices and edges in N G ¯ 1 , N G ¯ 2 . We prove that there is a total proper path between any two distinct vertices u and v in G . Note that P = x 1 x 3 x x 2 is a total proper path, where x i ∈ N G ¯ i . By means of the path P , we know that u and v are connected by some total proper path for any u ∈ N G ¯ i , v ∈ N G ¯ i + 1 . For any two vertices u , v ∈ N G ¯ 3 , it is trivial if u v ∈ E ( G ) . If u v ∉ E ( G ) , since u , v ∈ N G ¯ 3 , there exist two vertices u ′ , v ′ ∈ N G ¯ 3 such that u u ′ , v v ′ ∈ E ( G ¯ ) . Since G ¯ is triangle-free, we can see that u ′ ≠ v ′ and v u ′ , u v ′ ∈ E ( G ) . Then P = u x u ′ v ia a total proper path. Hence, G is total proper connected with the above coloring, and so tpc ( G ) = 3 .

Subcase 2.2. n 3 = 1 . Since G ¯ is triangle-free, we know that N G ¯ 1 is a stable set in G ¯ , and so a clique in G . Define a total coloring of G as follows: assign color 3 to the vertex x , and all edges between N G ¯ 1 and N G ¯ 3 ; assign color 2 to the edges x u for u ∈ N G ¯ 2 , and all vertices and edges in N G ¯ 3 ; assign color 1 to the edges x u for u ∈ N G ¯ 3 , and all vertices and edges in N G ¯ 1 , N G ¯ 2 . We prove that there is a total proper path between any two distinct vertices u and v in G . Note that P = x 1 x 3 x x 2 is a total proper path, where x i ∈ N G ¯ i . By means of the path P , we obtain that u and v are connected by some total proper path for any u ∈ N G ¯ i , v ∈ N G ¯ i + 1 . Let u , v be any two distinct vertices of N G ¯ 2 , and N G ¯ 3 = { y } . If y is adjacent to any vertex of N G ¯ 2 in G ¯ , then N G ¯ 2 is a clique in G , and so G has a Hamiltonian path. Otherwise, let V y denote the set of neighbors of y in N G ¯ 2 in G . We can check that P = u y x v is a total proper path, where u , v ∈ V y . If ∣ N G ¯ 2 \ V y ∣ = 1 , then P = u y x v is a total proper path, where u ∈ V y , v ∈ N G ¯ 2 \ V y . If ∣ N G ¯ 2 \ V y ∣ ≥ 2 , then G is claw-free since G ¯ is triangle-free, and G [ x ∪ N G ¯ 2 \ V y ] is a complete graph. Note that P = u y x v is a total proper path, where u ∈ V y , v ∈ N G ¯ 2 . Thus, G is total proper connected with the above coloring, and so tpc ( G ) = 3 .

Case 3. n 1 , n 2 , n 3 ≥ 2 . Since G ¯ is triangle-free, we have that N G ¯ 1 is a stable set in G ¯ , and so a clique in G . If any vertex in N G ¯ 3 is adjacent to all vertices of N G ¯ 2 in G ¯ , then both N G ¯ 2 and N G ¯ 3 are stable sets in G ¯ , and so cliques in G . Thus, G has a Hamiltonian path, and so tpc ( G ) = 3 .

Otherwise, we choose a vertex u ∈ N G ¯ 3 , let V u denote the set of neighbors of u in N G ¯ 2 in G , we have V u ≠ ∅ , N G ¯ 2 . Define a total coloring of G : assign color 2 to the vertex x , all vertices and edges in N G ¯ 1 , and all edges between N G ¯ 2 , N G ¯ 3 ; assign color 3 to the vertex u , all edges between N G ¯ 1 and N G ¯ 3 \ { u } , and all edges between x and V u ; assign color 2 to the remaining vertices and edges. Note that P = x v u x 1 is a total proper path, where v ∈ V u , x 1 ∈ N G ¯ 1 . For any two vertices w , z ∈ N G ¯ 3 , P = w x 1 u v x z is a total proper path, where x 1 ∈ N G ¯ 1 , v ∈ V u . For any two vertices w , z ∈ N G ¯ 2 , P = w u x v is a total proper path, where u , v ∈ V y . If ∣ N G ¯ 2 \ V y ∣ = 1 , then P = u x v is a total proper path, where u ∈ V y , v ∈ N G ¯ 2 \ V y . If ∣ N G ¯ 2 \ V y ∣ ≥ 2 , since G ¯ is triangle-free, we know that G is claw-free, and the subgraph G [ x ∪ N G ¯ 2 \ V y ] is a complete graph. Note that P = u x v is a total proper path, where u ∈ V y , v ∈ N G ¯ 2 \ V y . For any w ∈ N G ¯ 2 , z ∈ N G ¯ 3 , P = w x v u x 1 z is a total proper path, where v ∈ V y , x 1 ∈ N G ¯ 1 . Similarly, there is a total proper path connecting any two vertices w ∈ N G ¯ 2 , z ∈ N G ¯ 1 . Hence, G is total proper connected, and so tpc ( G ) = 3 .□

Lemma 3.3

For a connected graph G , if G ¯ is triangle-free and diam ( G ¯ ) = 2 , then tpc ( G ) = 3 .

Proof

Choose a vertex x with ecc G ¯ ( x ) = diam ( G ¯ ) = 2 . Since G is connected, we have n 1 ≥ 2 , n 2 = 1 or n 1 , n 2 ≥ 2 , and there exist two vertices u ∈ N G ¯ 1 , v ∈ N G ¯ 2 such that u v ∈ E ( G ) . Assume n 1 ≥ 2 and n 2 = 1 . Since G ¯ is triangle-free, we know that N G ¯ 1 is a stable set in G ¯ , and so a clique in G . Note that G has a Hamiltonian path, and so tpc ( G ) = 3 .

Assume n 1 , n 2 ≥ 2 . Observe that N G ¯ 1 is a stable set in G ¯ since G ¯ is triangle-free, and so a clique in G . We show a total coloring of G as follows: assign color 1 to the vertex x , the edge u v and all vertices in N G ¯ 1 \ u ; assign color 3 to the vertex v and all edges in N G ¯ 1 ; assign color 2 to the remaining vertices and edges. If there exist some vertices w ∈ N G ¯ 2 with d G ¯ ( w ) = n − 2 , then w is adjacent to the remaining vertices except x in G ¯ . Since diam ( G ¯ ) = 2 , there exists an edge w 1 w 2 ∈ E ( G ¯ ) with w 1 ∈ N G ¯ 1 , w 2 ∈ N G ¯ 2 . Thus, w , w 1 , w 2 is a triangle in G ¯ , a contradiction. Hence, d G ¯ ( w ) < n − 2 for all w ∈ N G ¯ 2 , and so d G ( w ) ≥ 2 . For any z ∈ N G ¯ 1 , we know that P = x v u z is a total proper path. For any y ∈ N G ¯ 2 \ { v } and z ∈ N G ¯ 1 , if N G ( y ) ∩ N G ¯ 1 ≠ ∅ , let w ∈ N G ( y ) ∩ N G ¯ 1 . Then y w z is a total proper path. Otherwise, let N G ( y ) ∩ N G ¯ 1 = ∅ . We claim that y is adjacent to all the other vertices of N G ¯ 2 in G . In fact, for any vertex w ∈ N G ¯ 2 \ { y } , there exists a vertex w ′ ∈ N G ¯ 1 such that w w ′ ∈ E ( G ¯ ) . Since y w ′ ∈ E ( G ¯ ) , we know that y w ∈ E ( G ) . Then y v u z is a total proper path. Next we consider w , z ∈ N G ¯ 2 such that w z ∉ E ( G ) . Since G ¯ is triangle-free, we have that G is claw-free, and at least one of w and z is adjacent to the v , without loss of generality, assume that w v ∈ E ( G ) . Since w , z ∈ N G ¯ 2 , there exist two vertices w ′ , z ′ ∈ N G ¯ 2 such that w w ′ , z z ′ ∈ E ( G ¯ ) , and w ′ ≠ z ′ . Then z w ′ , w z ′ ∈ E ( G ) and P = w v u w ′ z is a total proper path. Thus, G is total proper connected with the above coloring. Hence, tpc ( G ) = 3 .□

Lemma 3.4

Let G be a connected graph of order n ≥ 3 . If G ¯ is disconnected and triangle-free, then tpc ( G ) = 3 .

Proof

Suppose G ¯ is triangle-free and contains two connected components one of which is trivial. Let G 1 ¯ and G 2 ¯ be the two components of G ¯ , where V ( G 1 ¯ ) = { u } . Then u is adjacent to any other vertex in G . We will consider two cases according to the value of δ , where δ is the minimum degree of G . If δ = 1 , let d ( v ) = δ . Since G ¯ is triangle-free, we know that G is claw-free, and the subgraph G [ V ( G ) \ { v } ] is a complete graph. Thus, G has a Hamiltonian path, and so tpc ( G ) = 3 . If δ ≥ 2 , let d ( v ) = δ , D = V ( G ) \ { u , v } , and V v be the set of neighbors of v in G . Now we define a total coloring of G as follows: assign color 1 to the vertex v and all the edges between u and V v ; assign color 3 to the vertex u and all the edges between v and V v ; assign color 2 to the remaining vertices and edges. Since G is claw-free, we can find that the subgraph G [ V ( G ) \ { v } ∪ V v ] is a complete graph, and P = v 1 u v v 2 is a total proper path, where v 1 , v 2 ∈ V v . For any w ∈ V v , z ∈ D \ V v , we obtain that P = w u z is a total proper path. Thus, G is total proper connected with the above coloring, and so tpc ( G ) = 3 . Suppose G ¯ contains at least three connected components or exactly two nontrivial components. Then we have tpc ( G ) = 3 by the similar proof of Theorem 1.1.□

Proof of Theorem 1.2

If G ¯ is connected, the result holds for the case diam ( G ¯ ) ≥ 4 by Lemma 3.1, the case diam ( G ¯ ) = 3 by Lemma 3.2, and the case diam ( G ¯ ) = 2 by Lemma 3.3. If G ¯ is disconnected, the result holds by Lemma 3.4.□

4 Proof of Theorem 1.3

Suppose F ≅ K 2 . Note that G has a Hamiltonian path, and thus tpc ( G ) = 3 . Next, we compute the total proper connection number of G by proving the following claim.

Claim 1. Let G be a graph obtained by adding two pendant vertices { u 1 , u 2 } to a vertex v 1 of a complete graph K t . Then tpc ( G ) = 3 .

Proof

Since G is not a complete graph, we have tpc ( G ) ≥ 3 . Now we only need to prove tpc ( G ) ≤ 3 by the following cases.

Case 1. t ≡ 0 ( mod 3 ) . Assign a total coloring c to G as follows: Let c ( u 1 v 1 ) = 1 , c ( u 2 v 1 ) = 3 ; c ( v 3 i + 1 ) = c ( v 3 i + 2 v 3 i + 3 ) = 2 , c ( v 3 i + 2 ) = c ( v 3 i + 3 v 3 i + 4 ) = 1 , and c ( v 3 i + 3 ) = c ( v 3 i + 1 u 3 i + 2 ) = 3 , where 0 ≤ i ≤ t 3 − 1 . Observe that P 1 = u 1 v 1 v 2 ⋯ v t − 1 v t and P 2 = u 2 v 1 v t v t − 1 ⋯ v 3 v 2 are two total proper paths.

Case 2. t ≡ 1 ( mod 3 ) . Assign a total coloring c to G as follows: Let c ( u 1 v 1 ) = c ( v t − 1 v 1 ) = 1 , c ( v t ) = 2 , c ( u 2 v 1 ) = c ( v t v 1 ) = 3 . Let i be an integer with 0 ≤ i ≤ t 3 − 1 , c ( v 3 i + 1 ) = c ( v 3 i + 2 v 3 i + 3 ) = 2 , c ( v 3 i + 2 ) = c ( v 3 i + 3 v 3 i + 4 ) = 1 , and c ( v 3 i + 3 ) = c ( v 3 i + 1 u 3 i + 2 ) = 3 . We can find that P 1 = u 1 v 1 v 2 ⋯ v t − 1 v t and P 2 = u 2 v 1 v t − 1 ⋯ v 3 v 2 v t are two total proper paths.

Case 3. t ≡ 2 ( mod 3 ) . Assign a total coloring c to G as follows: Let c ( u 1 v 1 ) = c ( v t v 1 ) = c ( v t ) = c ( v t − 2 v 1 ) = 1 , c ( v t − 1 ) = 2 , c ( u 2 v 1 ) = c ( v t v t − 1 ) = c ( v t − 1 v 2 ) = 3 . Let i be an integer with 0 ≤ i ≤ t 3 − 1 , c ( v 3 i + 1 ) = c ( v 3 i + 2 v 3 i + 3 ) = 2 , c ( v 3 i + 2 ) = c ( v 3 i + 3 v 3 i + 4 ) = 1 , and c ( v 3 i + 3 ) = c ( v 3 i + 1 u 3 i + 2 ) = 3 . We can easily verify that P 1 = u 1 v 1 v 2 ⋯ v t − 1 v t , P 2 = u 2 v 1 v t − 2 v t − 3 ⋯ v 3 v 2 v t − 1 , and P 3 = v t v 1 u 2 are three total proper paths.

Thus, G is total proper connected with the above coloring, and so tpc ( G ) = 3 . This completes the proof of Claim 1.□

Suppose F ≅ 2 K 1 . Assume that N X ( u 1 ) ∩ N X ( u 2 ) = ∅ . Then G has a Hamiltonian path, and so tpc ( G ) = 3 . Otherwise, if ∣ N X ( u 1 ) ∩ N X ( u 2 ) ∣ = d X ( u 1 ) = d X ( u 2 ) = 1 , then we know that tpc ( G ) = 3 from Claim 1. If ∣ N X ( u 1 ) ∩ N X ( u 2 ) ∣ ≥ 2 , or ∣ N X ( u 1 ) ∩ N X ( u 2 ) ∣ = 1 , and max { d X ( u 1 ) , d X ( u 2 ) } ≥ 2 , then we can find that G has a Hamiltonian path. Thus, tpc ( G ) = 3 .

5 Proof of Theorem 1.4

Suppose F ≅ K 3 or P 3 . Note that G has a Hamiltonian path, and so tpc ( G ) = 3 . The following three claims will be used later.

Claim 2. Let G be a graph obtained by adding a pendant vertex u 3 adjacent to vertex u 1 or u 2 of graph in Claim 1. Then tpc ( G ) = 3 .

Proof

Without loss of generality, assume that u 3 is adjacent to u 1 . Let c ( u 1 ) = { 1 , 2 , 3 } \ { c ( u 1 v 1 ) , c ( v 1 ) } , c ( u 1 u 3 ) = c ( v 1 ) , and the remaining vertices and edges are assigned the same color as Claim 1. We can verify that G is total proper connected with the above coloring. Then tpc ( G ) = 3 . This completes the proof of Claim 2.□

Claim 3. Let G be a graph obtained by adding three vertices { u 1 , u 2 , u 3 } to a complete graph K t such that d ( u 1 ) = d ( u 3 ) = 1 , d ( u 2 ) = 2 , N ( u 1 ) ∩ N ( u 3 ) = ∅ , and ∣ N ( u 2 ) ∩ N ( u i ) ∣ = 1 , where 1 ≤ i ≤ 2 . Then tpc ( G ) = 3 .

Proof

Without loss of generality, assume that N ( u 2 ) ∩ N ( u 1 ) = v 1 , N ( u 2 ) ∩ N ( u 3 ) = v i . Let c ( u 3 v i ) = c ( v i v i + 1 ) , c ( v 1 u 2 ) = c ( v i v i − 1 ) , and the remaining vertices and edges are assigned the same color as Claim 1. Suppose t ≡ 0 ( mod 3 ) . Note that P 1 = u 1 v 1 v 2 ⋯ v t − 1 v t , P 2 = u 2 v 1 v t v t − 1 ⋯ v 3 v 2 , P 3 = u 3 v i u 2 , P 4 = u 1 v 1 v 2 ⋯ v i u 3 , and P 5 = u 3 v i v i − 1 ⋯ v 1 v t v t − 1 ⋯ v i + 1 are five total proper paths. Suppose t ≡ 1 ( mod 3 ) . Note that P 1 = u 1 v 1 v 2 ⋯ v t − 1 v t , P 2 = u 2 v 1 v t − 1 ⋯ v 3 v 2 v t , P 3 = u 3 v i u 2 , P 4 = u 1 v 1 v 2 ⋯ v i u 3 and P 5 = u 3 v i v i − 1 ⋯ v 2 v t v t − 1 ⋯ v i + 1 are five total proper paths. Suppose t ≡ 2 ( mod 3 ) . We can find that P 1 = u 1 v 1 v 2 ⋯ v t − 1 v t , P 2 = u 2 v 1 v t − 2 ⋯ v 3 v 2 v t − 1 , P 3 = u 3 v i u 2 , P 4 = u 1 v 1 v 2 ⋯ v i u 3 , P 5 = u 3 v i v i − 1 ⋯ v 2 v t − 1 v t − 2 ⋯ v i + 1 , and P 6 = u 3 v i v i − 1 v t are six total proper paths. Therefore, G is total proper connected with the above coloring, and so tpc ( G ) = 3 . This completes the proof of Claim 3.□

Claim 4. Let G be a graph obtained by adding a vertex u to the graph in Claim 1 such that d ( u ) = 2 and v 1 is adjacent to u . Then tpc ( G ) = 3 .

Proof

Since d ( u ) = 2 , without loss of generality, we assume that v i is adjacent to u . Let c ( u v 1 ) = 1 , c ( u v i ) = { 1 , 2 , 3 } \ { c ( v i ) , c ( v i v i − 1 ) } , and the remaining vertices and edges are assigned the same color as Claim 1. Suppose t ≡ 0 ( mod 3 ) . Note that P 1 = u v i v i − 1 ⋯ v 1 v t v t − 1 ⋯ v i + 1 , P 2 = u v i v i − 1 ⋯ v 1 u 1 and P 3 = u v 1 u 2 are three total proper paths. Suppose t ≡ 1 ( mod 3 ) . Note that P 1 = u v i v i − 1 ⋯ v 1 u 1 , P 2 = u v i v i − 1 ⋯ v 2 v t v t − 1 ⋯ v i + 1 , and P 3 = u v 1 u 2 are three total proper paths. Suppose t ≡ 2 ( mod 3 ) . We can find that P 1 = u v i v i − 1 ⋯ v 1 u 1 , P 2 = u v i v i − 1 ⋯ v 2 v t − 1 v t − 2 ⋯ v i + 1 , P 3 = u v 1 u 2 , and P 4 = u v i v i − 1 ⋯ v 1 v t are four total proper paths. Hence, G is total proper connected with the above coloring, and so tpc ( G ) = 3 . This completes the proof of Claim 4.□

Suppose F ≅ K 2 + K 1 . Let V ( K 2 ) = { u 1 , u 2 } and V ( K 1 ) = { u 3 } . Since diam ( G ) = 2 , we have N X ( u 1 ) ∩ N X ( u 2 ) ∩ N X ( u 3 ) = { v } , and so tpc ( G ) = 3 by Claim 2. Suppose F ≅ 3 K 1 . Assume N X ( u 1 ) ∩ N X ( u 2 ) ∩ N X ( u 3 ) = ∅ . Then tpc ( G ) = 3 by Claim 3. Assume N X ( u 1 ) ∩ N X ( u 2 ) ∩ N X ( u 3 ) ≠ ∅ . If d X ( u 1 ) = d X ( u 2 ) = d X ( u 3 ) = 1 , then tpc ( G ) ≥ 4 by [26, Proposition 2]. Define a total coloring of G as follows: c ( u 3 v ) = 4 with v ∈ N X ( u 3 ) , and the remaining vertices and edges are assigned the same color as Claim 1. We check that any two vertices have a total proper path, and so tpc ( G ) = 4 . Otherwise, we have d X ( u 1 ) + d X ( u 2 ) + d X ( u 3 ) ≥ 4 . Without loss of generality, let d X ( u 1 ) ≥ 2 , and u ∈ X \ { v } where v ∈ N X ( u 1 ) ∩ N X ( u 2 ) ∩ N X ( u 3 ) . Thus, tpc ( G ) = 3 by Claim 4.

6 Proof of Theorem 1.5

Case 1. diam ( G ) = 3 . We prove Case 1 by analyzing the structure of F .

Subcase 1.1. F ≅ K 3 or P 3 . Note that G has a Hamiltonian path, and so tpc ( G ) = 3 .

Subcase 1.2. F ≅ K 2 + K 1 . Denote V ( K 2 ) = { u 1 , u 2 } and V ( K 1 ) = { u 3 } . Suppose N X ( u 1 ) ∩ N X ( u 3 ) ≠ ∅ or N X ( u 2 ) ∩ N X ( u 3 ) ≠ ∅ . Without of loss generality, we may assume that N X ( u 1 ) ∩ N X ( u 3 ) ≠ ∅ . Then tpc ( G ) = 3 by Claim 2. Suppose N X ( u 1 ) ∩ N X ( u 3 ) = ∅ and N X ( u 2 ) ∩ N X ( u 3 ) = ∅ . Since diam ( G ) = 3 , we have N X ( u 1 ) ≠ ∅ and N X ( u 3 ) ≠ ∅ . Then G has a Hamiltonian path, and so tpc ( G ) = 3 .

Subcase 1.3. F ≅ 3 K 1 . Let V ( F ) = { u 1 , u 2 , u 3 } . Since diam ( G ) = 3 , we have N X ( u 1 ) ∩ N X ( u 2 ) ∩ N X ( u 3 ) = ∅ . Suppose there exists two vertices u i , u j ∈ V ( F ) satisfy N X ( u i ) ∩ N X ( u j ) ≠ ∅ . Without loss of generality, let u 1 and u 2 satisfy N X ( u 1 ) ∩ N X ( u 2 ) ≠ ∅ and v 1 ∈ N X ( u 1 ) ∩ N X ( u 2 ) . Assume d X ( u 1 ) = d X ( u 2 ) = 1 and v i ∈ N ( u 3 ) . Since G is not complete, we have tpc ( G ) ≥ 3 . To the contrary, suppose there exists a total coloring c of G using three colors such that G is total proper connected. Since any two vertices of G are connected by a total proper path, we have c ( u 1 v 1 ) ≠ c ( v 1 ) ≠ c ( u 2 v 1 ) . Without loss of generality, let c ( u 1 v 1 ) = 1 , c ( v 1 ) = 2 and c ( u 2 v 1 ) = 3 . Consider the total proper path P between u 1 and u , then the color of vertices and edges in P follows the sequence 1 , 2 , 3 , … , 1 , 2 , 3 , … . Thus, the value of ( c ( v i ) , c ( v i u ) ) is ( 1 , 2 ) , ( 2 , 3 ) , or ( 3 , 1 ) . Consider the total proper path Q between u 2 and u , then the color of vertices and edges in Q follows the sequence 3 , 2 , 1 , … , 3 , 2 , 1 , … . But the value of ( c ( v i ) , c ( v i u ) ) is ( 3 , 2 ) , ( 2 , 1 ) , or ( 1 , 3 ) , a contradiction. Assign a total coloring c to G as follows: c ( u 1 v 1 ) = 1 , c ( v 1 ) = c ( v i u ) = 2 , c ( u 2 v 1 ) = 3 , assign 4 to the remaining edges, and assign 1 to the remaining vertices. We can check that G is total proper connected with the above coloring, and so tpc ( G ) = 4 . Assume d X ( u 1 ) + d X ( u 2 ) ≥ 3 , without loss of generality, let d X ( u 1 ) ≥ 2 . If d X ( u 3 ) = 1 and N X ( u 1 ) ∩ N X ( u 3 ) ≠ ∅ , then we have tpc ( G ) = 3 by Claim 3; if N X ( u 1 ) ∩ N X ( u 3 ) = ∅ , or N X ( u 1 ) ∩ N X ( u 3 ) ≠ ∅ and d X ( u 3 ) ≥ 2 , then G has a Hamiltonian path, and so tpc ( G ) = 3 .

Now, we may suppose N X ( u 1 ) ∩ N X ( u 2 ) = ∅ , N X ( u 1 ) ∩ N X ( u 3 ) = ∅ , and N X ( u 2 ) ∩ N X ( u 3 ) = ∅ . Since G is not complete, we have tpc ( G ) ≥ 3 . To the contrary, assume that v i ∈ N ( u i ) , and there exists a total coloring c of G using three colors such that G is total proper connected. Since any two vertices of G are connected by a total proper path, we have c ( u 1 v 1 ) ≠ c ( v 1 ) . Without loss of generality, let c ( u 1 v 1 ) = 1 and c ( v 1 ) = 2 . Consider the total proper path P between u 1 and u 2 , then the color of vertices and edges in P follows the sequence 1 , 2 , 3 , … , 1 , 2 , 3 , … . Thus, the value of ( c ( v 2 ) , c ( v 2 u 2 ) ) is ( 1 , 2 ) , ( 2 , 3 ) , or ( 3 , 1 ) . Consider the total proper path Q between u 1 and u 3 , then the color of vertices and edges in Q follows the sequence 1 , 2 , 3 , … , 1 , 2 , 3 , … . Hence, the value of ( c ( v 3 ) , c ( v 3 u 3 ) ) is ( 1 , 2 ) , ( 2 , 3 ) , or ( 3 , 1 ) . Consider the total proper path W between u 2 and u 3 . If c ( v 2 ) = 1 and c ( v 2 u 2 ) = 2 , then the color of vertices and edges in W follows the sequence 2 , 1 , 3 , … , 2 , 1 , 3 , … . Note that the value of ( c ( v 3 ) , c ( v 3 u 3 ) ) is ( 2 , 1 ) , ( 1 , 3 ) , or ( 3 , 2 ) , a contradiction. If c ( v 2 ) = 2 and c ( v 2 u 2 ) = 3 , then the color of vertices and edges in W follows the sequence 3 , 2 , 1 , … , 3 , 2 , 1 , … . Note that the value of ( c ( v 3 ) , c ( v 3 u 3 ) ) is ( 2 , 1 ) , ( 1 , 3 ) , or ( 3 , 2 ) , a contradiction. If c ( v 2 ) = 3 and c ( v 2 u 2 ) = 1 , then the color of vertices and edges in W follows the sequence 1 , 3 , 2 , … , 1 , 3 , 2 , … . Note that the value of ( c ( v 3 ) , c ( v 3 u 3 ) ) is ( 2 , 1 ) , ( 1 , 3 ) , or ( 3 , 2 ) , a contradiction. Assign a total coloring c to G as follows: c ( u 1 v 1 ) = c ( v 2 ) = 1 , c ( v 1 ) = c ( u 2 v 2 ) = c ( u 3 v 3 ) = 2 , assign 4 to the remaining vertices, and assign 3 to the remaining edges. We can verify that G is total proper connected with the above coloring, and so tpc ( G ) = 4 .

Case 2. diam ( G ) ≥ 4 . Thus, F ≅ P 3 or F ≅ K 2 + K 1 . Assume F ≅ P 3 . Obviously, G has a Hamiltonian path, and so tpc ( G ) = 3 . Assume F ≅ K 2 + K 1 . Denote V ( K 2 ) = { u 1 , u 2 } and V ( K 1 ) = { u 3 } , without loss of generality, we have d X ( u 2 ) = 0 , d X ( u 1 ) ≥ 1 , d X ( u 3 ) ≥ 1 satisfying N X ( u 1 ) ∩ N X ( u 2 ) = ∅ . Hence, we can find that G has a Hamiltonian path, and so tpc ( G ) = 3 .

7 Proof of Theorem 1.6

The proof of Theorem 1.6 follows from the next two lemmas. First, we shall determine the total proper k -connection numbers of the circular ladders.

Lemma 7.1

Let n be an integer with n ≥ 3 . Then tpc ( CL 2 n ) = tpc 2 ( CL 2 n ) = 3 , tpc 3 ( CL 2 n ) = 4 .

Proof

Let n be an integer with n ≥ 3 . Since CL 2 n contains a Hamiltonian path that is not complete, we have tpc ( CL 2 n ) = 3 . Since tpc 2 ( CL 2 n ) ≥ tpc ( CL 2 n ) = 3 , we only need to prove tpc 2 ( CL 2 n ) ≤ 3 .

Case 1. n ≡ 0 ( mod 3 ) . Let n = 3 t . Assign a total coloring c to CL 2 n as follows: Let i be an integer with 0 ≤ i ≤ t − 1 , c ( u 3 i + 1 ) = c ( u 3 i + 2 u 3 i + 3 ) = c ( v 3 i + 2 ) = c ( v 3 i + 3 v 3 i + 4 ) = 1 , c ( u 3 i + 2 ) = c ( u 3 i + 3 u 3 i + 4 ) = c ( v 3 i + 3 ) = c ( v 3 i + 1 v 3 i + 2 ) = 2 , and c ( u 3 i + 3 ) = c ( u 3 i + 1 u 3 i + 2 ) = c ( v 3 i + 1 ) = c ( v 3 i + 2 v 3 i + 3 ) = 3 ; c ( u i v i ) , c ( u i ) , c ( v i ) ∈ { 1 , 2 , 3 } with c ( u i v i ) ≠ c ( u i ) ≠ c ( v i ) for 1 ≤ i ≤ n . Let x and y be any two distinct vertices of CL 2 n . By symmetry, we may assume that x = u 1 . If y = u i for 2 ≤ i ≤ n , then x u 2 u 3 ⋯ u i − 1 y and x u n u n − 1 ⋯ u i + 1 y are two total proper paths connecting x and y . If y = v 1 , then x y and x u n u n − 1 ⋯ u 2 y are two total proper paths connecting x and y . If y = v i , then x v 1 v n v n − 1 ⋯ v i + 1 y and x u n u n − 1 ⋯ u i y are two total proper paths connecting x and y . Thus, CL 2 n is total proper 2-connected with the above coloring.

Case 2. n ≡ 1 ( mod 3 ) . Let n = 3 t + 1 . Define a total coloring c of CL 2 n as follows: Let i be an integer with 0 ≤ i ≤ t − 1 , c ( u 3 i + 1 ) = c ( u 3 i + 2 u 3 i + 3 ) = c ( v 3 i + 3 ) = c ( v 3 i + 1 v 3 i + 2 ) = 1 , c ( u 3 i + 2 ) = c ( u 3 i + 3 u 3 i + 4 ) = c ( v 3 i + 2 ) = c ( v 3 i + 3 v 3 i + 4 ) = 3 , and c ( u 3 i + 3 ) = c ( u 3 i + 1 u 3 i + 2 ) = c ( v 3 i + 1 ) = c ( v 3 i + 2 v 3 i + 3 ) = 2 ; c ( u n ) = c ( v 1 v n ) = 1 , c ( u n v n ) = 2 , c ( v n ) = c ( u 1 u n ) = 3 , and c ( u j v j ) = 3 for 1 ≤ j ≤ n − 1 . Let x and y be any two distinct vertices of CL 2 n . We may assume that x = u i for 1 ≤ i ≤ n − 1 . If y = u j for i ≤ j ≤ n − 1 , then x u i + 1 u i + 2 ⋯ u j − 1 y and x u i − 1 u i − 2 ⋯ u 1 v 1 v 2 ⋯ v n − 1 u n − 1 u n − 2 ⋯ u j + 1 y are two total proper paths connecting x and y . If y = v j for 1 ≤ j ≤ n − 1 , then x u i − 1 u i − 2 ⋯ u 1 v 1 v 2 ⋯ v j − 1 y and x u i + 1 ⋯ u n − 1 v n − 1 v n − 2 ⋯ v j + 1 y are two total proper paths connecting x and y . If y = u n , then x u i + 1 ⋯ u n − 1 y and x u i − 1 u i − 2 ⋯ u 1 y are two total proper paths connecting x and y . If y = v n , then x u i + 1 ⋯ u n − 1 u n y and x u i − 1 u i − 2 ⋯ u 1 v 1 v 2 ⋯ v n − 1 y are two total proper paths connecting x and y . Assume that x = u n . If y = v j for 1 ≤ j ≤ n − 1 , then x u 1 u 2 ⋯ u n − 1 v n − 1 v n − 2 ⋯ v j + 1 y and x u n v n v 1 v 2 ⋯ v j − 1 y are two total proper paths connecting x and y . If y = v n , then x y and x u n − 1 u n − 2 ⋯ u 1 v 1 y are two total proper paths connecting x and y . Thus, CL 2 n is total proper 2-connected with the above coloring.

Case 3. n ≡ 2 ( mod 3 ) . Let n = 3 t + 2 . Define a total coloring c of CL 2 n as follows: Let i be an integer with 0 ≤ i ≤ t − 2 , c ( u 3 i + 1 ) = c ( u 3 i + 2 u 3 i + 3 ) = c ( v 3 i + 1 ) = c ( v 3 i + 2 v 3 i + 3 ) = 1 , c ( u 3 i + 2 ) = c ( u 3 i + 3 u 3 i + 4 ) = c ( v 3 i + 2 ) = c ( v 3 i + 3 v 3 i + 4 ) = 3 , and c ( u 3 i + 3 ) = c ( u 3 i + 1 u 3 i + 2 ) = c ( v 3 i + 1 ) = c ( v 3 i + 2 v 3 i + 3 ) = 2 ; c ( u n ) = c ( u n − 4 ) = c ( v n − 2 ) = c ( v n ) = 1 , c ( u n − 1 ) = c ( v n − 1 ) = c ( u n − 3 ) = c ( v n − 3 ) = 3 , c ( u n − 2 ) = c ( v n − 4 ) = 2 , c ( u n − 4 u n − 3 ) = c ( u n − 1 u n ) = c ( v n − 1 v n ) = c ( v n − 3 v n − 2 ) = c ( v n − 2 v n − 1 ) = 2 , c ( u n − 3 u n − 2 ) = c ( u n − 2 u n − 1 ) = c ( v n − 4 v n − 3 ) = c ( u 2 v 2 ) = 1 , c ( u 1 u 1 ) = c ( u 1 u n ) = c ( v 1 v n ) = 3 ; c ( u 4 j v 4 j ) = 1 , c ( u 4 j − 1 v 4 j − 1 ) = 2 , c ( u 4 j + 1 v 4 j + 1 ) = 3 , where 1 ≤ j ≤ t − 1 . Note that u 1 u 2 ⋯ u n − 2 v n − 2 v n − 3 ⋯ v 1 u 1 is a total proper cycle, any two distinct vertices of the cycle have two disjoint total proper paths. Now, we may assume that x = u i for 1 ≤ i ≤ n − 2 . If y = u n − 1 , then x u i + 1 ⋯ u n − 2 v n − 2 v n − 1 y and x u i − 1 ⋯ u 1 u n y are two total proper paths connecting x and y . If y = u n , then x u i + 1 ⋯ u n − 2 v n − 2 v n − 1 ⋯ v 1 v n y and x u i − 1 ⋯ u 1 v 1 v n y are two total proper paths connecting x and y . If y = v n − 1 , then x u i − 1 ⋯ u 3 v 3 v 2 v 1 v n y and x u i + 1 ⋯ u n − 2 v n − 2 y are two total proper paths connecting x and y . If y = v n , then x u i − 1 ⋯ u 1 u n y and x u i + 1 ⋯ u n − 2 v n − 2 v n − 1 y are two total proper paths connecting x and y . Assume that x = v i for 1 ≤ i ≤ n − 2 . If y = v n − 1 , then x v i + 1 ⋯ v n − 2 u n − 2 u n − 1 ⋯ u 1 u n u n − 1 y and x v i − 1 ⋯ v 1 v n y are two total proper paths connecting x and y . If y = v n , then x v i + 1 ⋯ v n − 2 u n − 2 u n − 1 u n y and x v i − 1 ⋯ v 1 y are two total proper paths connecting x and y . Thus, CL 2 n is total proper 2-connected with the above coloring.

To the contrary, suppose there exists a total proper 3-connected coloring c of CL 2 n using three colors. Considering u 1 and v 2 , u 1 u 2 v 2 , u 1 v 1 v 2 , and u 1 u n v n v n − 1 ⋯ v 3 v 2 must be three total proper paths connecting u 1 and v 2 . Then c ( u 1 u 2 ) ≠ c ( u 2 v 2 ) ≠ c ( u 2 ) . Considering u 1 and u 3 , u 1 u 2 u 3 , u 1 u n u n − 1 ⋯ u 4 u 3 , and u 1 v 1 v 2 v 3 u 3 must be three total proper paths connecting u 1 and u 3 . Hence, c ( u 1 u 2 ) ≠ c ( u 2 u 3 ) ≠ c ( u 2 ) , and so c ( u 2 v 2 ) = c ( u 2 u 3 ) . But then, there is no set of three disjoint total proper paths connecting u 3 and v 2 , a contradiction. Hence, tpc 3 ( CL 2 n ) ≥ 4 . Now we only need to prove tpc 3 ( CL 2 n ) ≤ 4 .

Case 1. n ≡ 0 ( mod 3 ) . Let n = 3 t . Assign a total coloring c to CL 2 n as follows: Let i be an integer with 0 ≤ i ≤ t − 1 , c ( u 3 i + 1 ) = c ( u 3 i + 2 u 3 i + 3 ) = c ( v 3 i + 2 ) = c ( v 3 i + 3 v 3 i + 4 ) = 1 , c ( u 3 i + 2 ) = c ( u 3 i + 3 u 3 i + 4 ) = c ( v 3 i + 3 ) = c ( v 3 i + 1 v 3 i + 2 ) = 2 , and c ( u 3 i + 3 ) = c ( u 3 i + 1 u 3 i + 2 ) = c ( v 3 i + 1 ) = c ( v 3 i + 2 v 3 i + 3 ) = 3 ; c ( u j v j ) = 4 for 1 ≤ j ≤ n . Let x and y be any two distinct vertices of CL 2 n . By symmetry, we may assume that x = u 1 . If y = u i for 2 ≤ i ≤ n , then x u 2 u 3 ⋯ u i − 1 y , x u n u n − 1 ⋯ u i + 1 y , and x v 1 v 2 ⋯ v i y are three total proper paths connecting x and y . If y = v 1 , then x y , x u 2 v 2 y and x u n v n y are three total proper paths connecting x and y . If y = v i for 2 ≤ i ≤ n , then x u 1 u 2 ⋯ u i y , x v 1 v 2 ⋯ v i − 1 y , and x u n v n v n − 1 ⋯ u i y are three total proper paths connecting x and y . Hence, CL 2 n is total proper 3-connected with the above coloring.

Case 2. n ≡ 1 ( mod 3 ) . Let n = 3 t + 1 . Define a total coloring c of CL 2 n as follows: c ( u 3 i + 1 ) = c ( u 3 i + 2 u 3 i + 3 ) = c ( v 3 i + 3 ) = c ( v 3 i + 1 v 3 i + 2 ) = 1 , c ( u 3 i + 2 ) = c ( u 3 i + 3 u 3 i + 4 ) = c ( v 3 i + 2 ) = c ( v 3 i + 3 v 3 i + 4 ) = 3 , and c ( u 3 i + 3 ) = c ( u 3 i + 1 u 3 i + 2 ) = c ( v 3 i + 1 ) = c ( v 3 i + 2 v 3 i + 3 ) = 2 , where 0 ≤ i ≤ t − 1 ; c ( u n ) = c ( v 1 v n ) = 1 , c ( u n v n ) = 2 , c ( v n ) = c ( u 1 u n ) = 4 , and c ( u j v j ) = 4 for 2 ≤ j ≤ n − 1 . Let x and y be any two distinct vertices of CL 2 n . By symmetry, we may assume that x = u 1 . If y = u i for 2 ≤ i ≤ n , then x u 2 u 3 ⋯ u i − 1 y , x u n u n − 1 ⋯ u i + 1 y , and x v 1 v 2 ⋯ v i y are three total proper paths connecting x and y . If y = v 1 , then x y , x u 2 v 2 y and x u n v n y are three total proper paths connecting x and y . If y = v i for 2 ≤ i ≤ n , then x u 1 u 2 ⋯ u i y , x v 1 v 2 ⋯ v i − 1 y , and x u n v n v n − 1 ⋯ u i y are three total proper paths connecting x and y . Hence, CL 2 n is total proper 3-connected with the above coloring.

Case 3. n ≡ 2 ( mod 3 ) . Let n = 3 t + 2 . Define a total coloring c of CL 2 n as follows: c ( u 3 i + 1 ) = c ( u 3 i + 2 u 3 i + 3 ) = c ( v 3 i + 4 ) = c ( v 3 i + 2 v 3 i + 3 ) = 1 , c ( u 3 i + 2 ) = c ( u 3 i + 3 u 3 i + 4 ) = c ( v 3 i + 3 ) = c ( v 3 i + 1 v 3 i + 2 ) = 3 , and c ( u 3 i + 3 ) = c ( u 3 i + 1 u 3 i + 2 ) = c ( v 3 i + 2 ) = c ( v 3 i + 3 v 3 i + 4 ) = 2 , where 0 ≤ i ≤ t − 1 ; c ( u n ) = c ( v n − 1 v n ) = 3 , c ( v n ) = c ( u n − 1 u n ) = c ( u 1 v 1 ) = 2 , c ( u n − 1 ) = c ( v 1 v n ) = c ( u n u 1 ) = 1 , c ( v 1 ) = 4 , and c ( u j v j ) = 4 for 2 ≤ j ≤ n . Let x and y be any two distinct vertices of CL 2 n . By symmetry, we may assume that x = u 1 . If y = u i for 2 ≤ i ≤ n , then x u 2 u 3 ⋯ u i − 1 y , x u n u n − 1 ⋯ u i + 1 y , and x v 1 v 2 ⋯ v i y are three total proper paths connecting x and y . If y = v 1 , then x y , x u 2 v 2 y , and x u n v n y are three total proper paths connecting x and y . If y = v i for 2 ≤ i ≤ n , then x u 1 u 2 ⋯ u i y , x v 1 v 2 ⋯ v i − 1 y and x u n v n v n − 1 ⋯ u i y are three total proper paths connecting x and y . Hence, CL 2 n is total proper 3-connected with the above coloring.□

Next, we shall determine the total proper k -connection numbers of the Möbius ladders.

Lemma 7.2

Let n be an integer with n ≥ 3 . Then tpc ( M 2 n ) = tpc 2 ( M 2 n ) = 3 , tpc 3 ( M 2 n ) = 4 .

Proof

Since M 2 n contains a Hamiltonian path and is not complete, we have tpc ( M 2 n ) = 3 . Since tpc 2 ( M 2 n ) ≥ tpc ( M 2 n ) = 3 , we only need to prove tpc 2 ( M 2 n ) ≤ 3 .

Case 1. n ≡ 0 ( mod 3 ) . Define a total coloring c of M 2 n as follows: Let i be an integer with 1 ≤ i ≤ n − 2 , c ( u i ) = c ( v n − i + 1 ) = 1 , c ( u i + 1 ) = c ( v n − i ) = 3 , and c ( u i + 2 ) = c ( v n − i − 1 ) = 2 ; c ( u i u i + 1 ) , c ( u i ) , c ( u i + 1 ) ∈ { 1 , 2 , 3 } with c ( u i u i + 1 ) ≠ c ( u i ) ≠ c ( u i + 1 ) for 1 ≤ i ≤ n − 1 ; c ( v i v i + 1 ) , c ( v i ) , c ( v i + 1 ) ∈ { 1 , 2 , 3 } with c ( v i v i + 1 ) ≠ c ( v i ) ≠ c ( v i + 1 ) for 1 ≤ i ≤ n − 1 ; c ( u 1 v 1 ) = 3 , c ( u n v n ) = 2 , and c ( u j v n − j + 1 ) = 3 for 1 ≤ j ≤ n . Let x and y be any two distinct vertices of M 2 n . By symmetry, we may assume that x = u 1 . If y = u i for 2 ≤ i ≤ n , then x u 2 u 3 ⋯ u i − 1 y and x v 1 v 2 ⋯ v n u n u n − 1 ⋯ u i + 1 y are two total proper paths connecting x and y . If y = v 1 , then x y and x u 2 u 3 y are two total proper paths connecting x and y . If y = v i for 2 ≤ i ≤ n , then x u 1 u 2 ⋯ u n v n v n − 1 ⋯ v i + 1 y and x v 1 v 2 ⋯ v i − 1 y are two total proper paths connecting x and y . Thus, M 2 n is total proper 2-connected with the above coloring.

Case 2. n ≡ 1 ( mod 3 ) . Let n = 3 t + 1 . Define a total coloring c of M 2 n as follows: Let i be an integer with 0 ≤ i ≤ t − 1 , c ( u 3 i + 1 ) = c ( u 3 i + 2 u 3 i + 3 ) = c ( v 3 i + 3 ) = c ( v 3 i + 1 v 3 i + 2 ) = 1 , c ( u 3 i + 2 ) = c ( u 3 i + 3 u 3 i + 4 ) = c ( v 3 i + 1 ) = c ( v 3 i + 2 v 3 i + 3 ) = 3 , and c ( u 3 i + 3 ) = c ( u 3 i + 1 u 3 i + 2 ) = c ( v 3 i + 2 ) = c ( v 3 i + 3 v 3 i + 4 ) = 2 ; if j = 3 i for 1 ≤ i ≤ t , then c ( u j v n − j + 1 ) = 3 ; if j ≠ 3 i for 1 ≤ i ≤ t , then c ( u j v n − j + 1 ) , c ( u j ) , c ( v n − j + 1 ) ∈ { 1 , 2 , 3 } with c ( u j v n − j + 1 ) ≠ c ( u j ) ≠ c ( v n − j + 1 ) ; c ( u n ) = 1 , c ( u n v n ) = c ( u 1 v 1 ) = 2 , c ( v n ) = 3 . Let x and y be any two distinct vertices of M 2 n . We may assume that x = u i for 1 ≤ i ≤ n . If y = u j for i ≤ j ≤ n , then x u i + 1 u i + 2 ⋯ u j − 1 y and x u i − 1 u i − 2 ⋯ u 1 v n u n u n − 1 ⋯ u j + 1 y are two total proper paths connecting x and y . If y = v j for 1 ≤ j ≤ n , then x u i − 1 u i − 2 ⋯ u 1 v n v n − 1 ⋯ v j + 1 y and x u i + 1 ⋯ u n − j + 1 y are two total proper paths connecting x and y . Thus, M 2 n is total proper 2-connected with the above coloring.

Case 3. n ≡ 2 ( mod 3 ) . Let n = 3 t + 2 . Define a total coloring c of M 2 n as follows: Let i be an integer with 0 ≤ i ≤ t − 1 , c ( u 3 i + 1 ) = c ( u 3 i + 2 u 3 i + 3 ) = c ( v 3 i + 3 ) = c ( v 3 i + 1 v 3 i + 2 ) = c ( u 3 i + 2 v n − 3 i − 1 ) = 1 , c ( u 3 i + 2 ) = c ( u 3 i + 3 u 3 i + 4 ) = c ( v 3 i + 1 ) = c ( v 3 i + 2 v 3 i + 3 ) = c ( u 3 i + 3 v n − 3 i − 2 ) = 3 , and c ( u 3 i + 3 ) = c ( u 3 i + 1 u 3 i + 2 ) = c ( v 3 i + 2 ) = c ( v 3 i + 3 v 3 i + 4 ) = c ( u 3 i + 4 v n − 3 i − 3 ) = 2 ; c ( u n − 1 ) = c ( u n v 1 ) = c ( v n − 1 v n ) = 1 , c ( u n ) = c ( v n − 1 ) = c ( u 1 v n ) = 3 , c ( v n ) = c ( u n − 1 u n ) = 2 . Let x and y be any two distinct vertices of M 2 n . We may assume that x = u i for 1 ≤ i ≤ n . If y = u j for i ≤ j ≤ n , then x u i + 1 u i + 2 ⋯ u j − 1 y and x u i − 1 u i − 2 ⋯ u 1 v n u n u n − 1 ⋯ u j + 1 y are two total proper paths connecting x and y . If y = v j for 1 ≤ j ≤ n , then x u i − 1 u i − 2 ⋯ u 1 v n v n − 1 ⋯ v j + 1 y and x u i + 1 ⋯ u n − j + 1 y are two total proper paths connecting x and y . Thus, M 2 n is total proper 2-connected with the above coloring.

To the contrary, suppose there exists a total proper 3-connected coloring c of M 2 n using three colors. By considering the pair { u 2 , v n } , u 2 u 3 ⋯ u n v n , u 2 u 1 v n , and u 2 v n − 1 v n must be three total proper paths connecting u 2 and v n . Then c ( u 2 v n − 1 ) ≠ c ( v n − 1 v n ) ≠ c ( v n − 1 ) . By considering the pair { u 2 , v n − 2 } , u 2 u 1 v n v n − 1 v n − 2 , u 2 u 3 v n − 2 , and u 2 v n − 1 v n must be three total proper paths connecting u 2 and v n − 2 . Thus, c ( u 2 v n − 1 ) ≠ c ( v n − 1 v n − 2 ) ≠ c ( v n − 1 ) , and hence c ( v n − 1 v n − 2 ) = c ( v n − 1 v n ) . But then, there is no set of three disjoint total proper paths connecting v n − 2 and v n , a contradiction. Thus, tpc 3 ( M 2 n ) ≥ 4 . Now we only need to prove tpc 3 ( M 2 n ) ≤ 4 .

Case 1. n ≡ 0 ( mod 2 ) . Let n = 2 t . Assign a total coloring c of M 2 n as follows: c ( u 2 i + 1 ) = c ( v n − 2 i ) = c ( u 2 i + 2 v n − 2 i − 1 ) = 1 , c ( u 2 i + 2 ) = c ( u n − 2 i − 1 ) = c ( u 2 i + 1 v n − 2 i ) = 3 , c ( v n − 2 i v n − 2 i − 1 ) = c ( u 2 i + 1 u 2 i + 2 ) = 2 , where 0 ≤ i ≤ t − 1 ; c ( u 2 i + 2 u 2 i + 3 ) = c ( v n − 2 i − 1 v n − 2 i − 2 ) = 4 for 0 ≤ i ≤ t − 2 ; c ( u n v n ) = c ( u 1 v 1 ) = 4 . Let x and y be any two distinct vertices of M 2 n . By symmetry, we may assume that x = u 1 . If y = u i for 2 ≤ i ≤ n , then x u 2 ⋯ u i − 1 y , x v n u n u n − 1 ⋯ u i + 1 y , and x v 1 v 2 ⋯ v n − i + 1 u i are three total proper paths connecting x and y . If y = v 1 , then x y , x u 2 u 3 ⋯ u n y , and x v n v n − 1 ⋯ v 2 y are three total proper paths connecting x and y . If y = v i for 2 ≤ i ≤ n , then x v 1 v 2 ⋯ v i − 1 y , x v n v n − 1 ⋯ v i + 1 y and x u 2 u 3 ⋯ u n − i + 1 y are three total proper paths connecting x and y . Hence, M 2 n is total proper 3-connected with the above coloring.

Case 2. n ≡ 1 ( mod 2 ) . For n = 3 , we assign a total coloring c to M 6 as follows: c ( u 1 ) = c ( v 3 ) = c ( u 2 u 3 ) = c ( v 1 v 2 ) = 1 , c ( u 2 ) = c ( v 2 ) = c ( u 3 v 3 ) = c ( u 1 v 1 ) = 2 , c ( u 3 ) = c ( v 1 ) = c ( u 1 u 2 ) = c ( v 2 v 3 ) = 3 , c ( u i v n − i + 1 ) = 4 for 1 ≤ i ≤ 3 . We can verify that M 6 is total proper 3-connected with the above total coloring, and so tpc 3 ( M 6 ) = 4 .

For n = 5 , define a total coloring c of M 10 as follows: c ( u 1 ) = c ( v 5 ) = 1 , c ( u 2 ) = c ( v 4 ) = 2 , c ( u 3 ) = c ( v 3 ) = c ( u 5 ) = c ( v 1 ) = 3 , c ( u 4 ) = c ( v 2 ) = 4 , c ( u 2 u 3 ) = c ( u 4 u 5 ) = c ( v 1 v 2 ) = c ( v 3 v 4 ) = 1 , c ( u 3 u 4 ) = c ( u 5 v 5 ) = c ( u 1 v 1 ) = c ( v 2 v 3 ) = 2 , c ( u 4 v 2 ) = c ( u 1 u 2 ) = c ( v 4 v 5 ) = 3 , c ( u i v n − i + 1 ) = 4 for i = 1 , 2 , 3 , 5 . We can check that M 10 is total proper 3-connected with the above total coloring, and so tpc 3 ( M 10 ) = 4 .

Subcase 2.1. Let n ≡ 1 ( mod 4 ) for n ≥ 7 . Let n = 2 t + 1 . Assign a total coloring c to M 2 n as follows: c ( u 4 i + 1 ) = c ( v n − 4 i ) = 1 , c ( u 4 i + 2 ) = c ( v n − 4 i − 1 ) = 2 , c ( u 4 i + 3 ) = c ( v n − 4 i − 2 ) = 3 , and c ( u 4 i + 4 ) = c ( v n − 4 i − 3 ) = 4 , where 0 ≤ i ≤ t − 2 2 ; c ( u 4 i u 4 i + 1 ) = c ( v n − 4 i + 1 v n − 4 i ) = c ( u 4 i + 2 v n − 4 i − 1 ) = 3 , c ( u 4 i + 1 u 4 i + 2 ) = c ( v n − 4 i v n − 4 i − 1 ) = c ( u 4 i + 3 v n − 4 i − 2 ) = 4 , c ( u 4 i + 2 u 4 i + 3 ) = c ( v n − 4 i − 1 v n − 4 i − 2 ) = c ( u 4 i v n − 4 i + 1 ) = 1 , and c ( v n − 4 i − 2 v n − 4 i − 3 ) = c ( u 4 i + 3 u 4 i + 4 ) = c ( u 4 i + 1 v n − 4 i ) = 2 , where 1 ≤ i ≤ t − 2 2 ; c ( u n ) = c ( v 1 ) = c ( u 1 u 2 ) = c ( v n v n − 1 ) = 3 , c ( u 2 u 3 ) = c ( v n − 1 v n − 2 ) = c ( v 1 v 2 ) = c ( u n − 1 u n ) = c ( u n − 1 v 2 ) = 1 , c ( u 3 u 4 ) = c ( v n − 2 v n − 3 ) = c ( u 1 v 1 ) = c ( u n v n ) = 2 , and c ( u 1 v n ) = c ( u 2 v n − 1 ) = c ( u 3 v n − 2 ) = 4 . Let x and y be any two distinct vertices of M 2 n . By symmetry, we may assume that x = u 1 . If y = u i for 2 ≤ i ≤ n , then x u 2 u 3 ⋯ u i − 1 y , x v n u n u n − 1 ⋯ u i + 1 y , and x v 1 v 2 ⋯ v n − i + 1 u i are three total proper paths connecting x and y . If y = v 1 , then x y , x u 2 u 3 ⋯ u n y , and x v n v n − 1 ⋯ v 2 y are three total proper paths connecting x and y . If y = v i for 2 ≤ i ≤ n , then x v 1 v 2 ⋯ v i − 1 y , x v n v n − 1 ⋯ v i + 1 y and x u 2 u 3 ⋯ u n − i + 1 y are three total proper paths connecting x and y . Hence, M 2 n is total proper 3-connected with the above coloring.

Subcase 2.2. Let n ≡ 3 ( mod 4 ) for n ≥ 7 . Let n = 2 t + 1 . Assign a total coloring c to M 2 n as follows: Let i be an integer with 0 ≤ i ≤ t − 3 2 , c ( u 4 i + 1 ) = c ( v n − 4 i ) = 1 , c ( u 4 i + 2 ) = c ( v n − 4 i − 1 ) = 2 , c ( u 4 i + 3 ) = c ( v n − 4 i − 2 ) = 3 , and c ( u 4 i + 4 ) = c ( v n − 4 i − 3 ) = 4 . Let i be an integer with 1 ≤ i ≤ t − 1 2 , c ( u 4 i u 4 i + 1 ) = c ( v n − 4 i − 1 v n − 4 i ) = c ( u 4 i + 2 v n − 4 i − 1 ) = 3 , c ( u 4 i + 1 u 4 i + 2 ) = c ( v n − 4 i v n − 4 i − 1 ) = c ( u 4 i + 3 v n − 4 i − 2 ) = 4 , c ( u 4 i + 2 u 4 i + 3 ) = c ( u 4 i v n − 4 i + 1 ) = c ( v n − 4 i − 1 v n − 4 i − 2 ) = 1 , and c ( u 4 i + 1 v n − 4 i ) = c ( u 4 i − 1 u 4 i ) = c ( v n − 4 i + 2 v n − 4 i + 1 ) = 2 ; c ( u n ) = c ( v 1 ) = c ( u 1 u 2 ) = c ( v n v n − 1 ) = 3 , c ( u n − 1 ) = c ( v 2 ) = c ( u 1 v 1 ) = c ( u n v n ) = 2 , and c ( u n − 2 ) = c ( v 3 ) = c ( u 2 u 3 ) = c ( v n − 1 v n − 2 ) = 1 . By the similar proof of the above subcase, we can verify M 2 n is total proper 3-connected with the above coloring.□

8 Proof of Theorem 1.7

Note that K 3 □ K 2 = CL 6 , K 3 , 3 = M 6 , Q 3 = CL 8 and M 8 . By means of Theorem 1.6, we can obtain their total proper k -connection numbers. Now, we only need to consider K 4 , F 1 , F 2 , F 3 .

Lemma 8.1

tpc ( K 4 ) = 1 , tpc 2 ( K 4 ) = 3 , tpc 3 ( K 4 ) = 4 .

Proof

By [26], we know that tpc ( K 4 ) = 1 . Suppose tpc 2 ( K 4 ) = 2 , then there is no set of two disjoint total proper paths connecting u 1 and u 2 , a contradiction. Thus, tpc 2 ( K 4 ) ≥ 3 . Let V ( K 4 ) = { u 1 , u 2 , u 3 , u 4 } , we assign a total coloring c to K 4 as follows: c ( u 1 ) = c ( u 2 u 3 ) = c ( u 3 u 4 ) = c ( u 2 u 4 ) = 1 , c ( u 2 ) = c ( u 4 ) = c ( u 1 u 3 ) = 2 , c ( u 3 ) = c ( u 1 u 2 ) = c ( u 1 u 4 ) = 3 . We can verify that the K 4 is total proper 2-connected, so tpc 2 ( K 4 ) = 3 .

Now, we suppose there exists a total proper 3-connected coloring c of K 4 using three colors. Considering u 1 and u 2 , u 1 u 2 , u 1 u 4 u 2 , and u 1 u 3 u 2 must be the three total proper paths connecting u 1 and u 2 . Then c ( u 1 u 4 ) ≠ c ( u 2 u 4 ) ≠ c ( u 4 ) . Considering u 1 and u 3 , u 1 u 3 , u 1 u 2 u 3 , and u 1 u 4 u 3 must be the three total proper paths connecting u 1 and u 3 . Thus, c ( u 1 u 4 ) ≠ c ( u 3 u 4 ) ≠ c ( u 4 ) , and so c ( u 2 u 4 ) = c ( u 4 u 3 ) . But then, there is no set of three disjoint total proper paths connecting u 2 and u 3 , a contradiction. Hence, tpc 3 ( K 4 ) ≥ 4 . Define a total coloring c of K 4 as follows: c ( u 1 ) = c ( u 3 ) = 1 , c ( u 2 ) = c ( u 4 ) = 2 , c ( u 1 u 2 ) = c ( u 3 u 4 ) = 4 , c ( u 1 u 4 ) = c ( u 2 u 3 ) = 3 , c ( u 1 u 3 ) = 2 , c ( u 2 u 4 ) = 1 . We can easily check that K 4 is total proper 3-connected, and so tpc 3 ( K 4 ) = 4 .□

Lemma 8.2

tpc ( F 1 ) = tpc 2 ( F 1 ) = 3 , tpc 3 ( F 1 ) = 4 .

Figure 4 The total proper k-connected coloring of F1, F2 and F3.
Figure 4

The total proper k-connected coloring of F1, F2 and F3.

Proof

Since F 1 has a Hamiltonian path that is not complete, we know that tpc ( F 1 ) = 3 . It is easy to verify that F 1 is total proper 2-connected depicted in Figure 4(a), and so tpc 2 ( F 1 ) = 3 . Now, we suppose there exists a total proper 3-connected coloring c of F 1 using three colors. By considering the pair { u 2 , u 8 } , u 2 u 8 , u 2 u 1 u 8 , and u 2 u 3 ⋯ u 8 must be the three total proper paths connecting u 2 and u 8 . Then c ( u 1 u 2 ) ≠ c ( u 1 u 8 ) ≠ c ( u 1 ) . By considering the pair { u 5 , u 8 } , u 5 u 6 u 7 u 8 , u 5 u 1 u 8 , and u 5 u 4 u 3 u 2 u 8 must be the three total proper paths connecting u 5 and u 8 . Then c ( u 1 u 5 ) ≠ c ( u 1 u 8 ) ≠ c ( u 1 ) , and hence c ( u 1 u 2 ) = c ( u 1 u 5 ) . But then, there is no set of three disjoint total proper paths connecting u 2 and u 5 , a contradiction. Hence, tpc 3 ( F 1 ) ≥ 4 . By Figure 4(b), we know that F 1 is total proper 3-connected, and so tpc 3 ( F 1 ) = 4 .□

Lemma 8.3

tpc ( F 2 ) = tpc 2 ( F 2 ) = 3 .

Proof

Since F 1 has a Hamiltonian path that is not complete, we know that tpc ( F 2 ) = 3 . We can check that the coloring shown in Figure 4(c) is total proper 2-connected. Thus, tpc 2 ( F 2 ) = 3 .□

Lemma 8.4

tpc ( F 3 ) = tpc 2 ( F 3 ) = 3 , tpc 3 ( F 3 ) = 4 .

Proof

Since F 3 has a Hamiltonian path that is not complete, we know that tpc ( F 3 ) = 3 . It is easy to check that the coloring shown in Figure 4(d) is total proper 2-connected using three colors. Thus, tpc 2 ( F 3 ) = 3 . Now, we suppose there exists a total proper 3-connected coloring c of F 3 using three colors. Considering u 2 and u 8 , u 2 u 8 , u 2 u 1 u 8 and u 2 u 3 ⋯ u 8 must be the three total proper paths connecting u 2 and u 8 . Then c ( u 1 u 2 ) ≠ c ( u 1 u 8 ) ≠ c ( u 1 ) . Considering u 5 and u 8 , u 5 u 6 u 7 u 8 , u 5 u 1 u 8 , and u 5 u 4 u 3 u 2 u 8 must be the three total proper paths connecting u 5 and u 8 . Thus, c ( u 1 u 5 ) ≠ c ( u 1 u 8 ) ≠ c ( u 1 ) , and so c ( u 1 u 2 ) = c ( u 1 u 5 ) . But then, there is no set of three disjoint total proper paths connecting u 2 and u 5 , a contradiction. Hence, tpc 3 ( F 3 ) ≥ 4 . By Figure 4(e), we know that F 3 is total proper 3-connected, and so tpc 3 ( F 3 ) = 4 .□

Acknowledgements

Special thanks should go to editor and reviewers who have put considerable time and effort into their comments on this paper.

  1. Funding information: This work was supported by the NSFC (No. 11701157 and No. 11901196), the Foundation of Henan Educational Committee (22A110003), and the Foundation of Henan Normal University (No. 2019QK06 and No. 2020PL05).

  2. Author contributions: Yingbin Ma and Hui Zhang contributed to the design and implementation of the research, to the analysis of the results, and to the writing of the manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2021-12-03
Revised: 2022-02-17
Accepted: 2022-02-20
Published Online: 2022-04-08

© 2022 Yingbin Ma and Hui Zhang, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0025
Schlagwörter für diesen Artikel
total coloring; total proper path; total proper k-connected; total proper k-connection number; complement graph; clique number
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Artikel in diesem Heft

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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