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On k-prime graphs

  • Omar A. Abughneim EMAIL logo and Baha’ Abughazaleh
Published/Copyright: December 6, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In the context of a simple undirected graph G , a k -prime labeling refers to assigning distinct integers from the set { k , k + 1 , … , ∣ V ( G ) ∣ + k − 1 } to its vertices, such that adjacent vertices in G are labeled with numbers that are relatively prime to each other. If G has a k -prime labeling, we say that G is a k -prime graph (k-PG). In this article, we characterize when a graph up to order 6 is a k-PG and characterize when a graph of order 7 is a k-PG whenever k and k + 1 are not divisible by 5. Also, we find a lower bound for the independence number of a k-PG. Finally, we study when a cycle is a k-PG.

Keywords: independence number; prime labeling; prime graphs; k-prime graphs; maximal k-prime graphs
MSC 2010: 05C78

1 Introduction

A set S of vertices of an undirected graph G is called an independent set if it has no adjacent pairs in G . The independence number of G is the maximum cardinality of an independent set of G , denoted by α ( G ) . A prime labeling (PL) of graph G is defined by a bijective map f from its vertex set to the set { 1 , 2 , … , ∣ V ( G ) ∣ } such that f ( u ) and f ( v ) are relatively prime for every pair of adjacent vertices u and v in G and we say that G is a prime graph (PG) if it has a PL. In [1], Tout et al. introduced a PL, a concept defined by Entringer. Various families of graphs such as paths, cycles, stars, and spiders are proven to be prime [2,3]. More results on PL can be found in [4–8]. Vaidya and Prajapati [9] generalized the definition of PL as follows. A k -prime labeling (k-PL) of graph G is defined by a bijective map f from its vertex set to the set { k , k + 1 , … , ∣ V ( G ) ∣ + k − 1 } such that f ( u ) and f ( v ) are relatively prime for every pair of adjacent vertices u and v in G . Moreover, G is called a k -prime graph (k-PG). If k = 1 , then a k-PL is the same as PL. In this article, we characterize when a graph up to order 6 is a k-PG and characterize when a graph of order 7 is a k -PG whenever k and k + 1 are not divisible by 5. Also, we establish a lower bound for the independence number of a k-PG. Finally, we studied when a cycle is a k-PG and we generalized and modified some results in [10]. For notions and terminology not explicitly defined here, please refer to [11–13].

2 k -prime graphs

It is clear that a spanning subgraph of a k-PG is a k-PG. We state this in the following lemma.

Lemma 2.1

If G is a k-PG, then every spanning subgraph of G is a k-PG.

Vaidya and Prajapati [9] proved the following two lemmas.

Lemma 2.2

Let G 1 be a PG of order n and G 2 be an ( n + 1 ) -PG. Then, G 1 ∪ G 2 is a PG.

Lemma 2.3

Any path P n is a k-PG for all k ∈ N .

By Lemma 2.3, it is obvious that every graph of order 1 or 2 is a k-PG for all k ∈ N .

In Lemma 2.4, we find a restriction on the independence number of a k-PG.

Lemma 2.4

If a graph G is a k-PG, then α ( G ) ≥ ∣ V ( G ) ∣ 2 if k is e v e n , ∣ V ( G ) ∣ 2 if k is o d d .

Proof

Let f be a k-PL of G . Let E be the set of all vertices of G whose labels are even. Then, E is an independent set with ∣ E ∣ = ∣ V ( G ) ∣ 2 if k is even , ∣ V ( G ) ∣ 2 if k is odd . Therefore, α ( G ) ≥ ∣ V ( G ) ∣ 2 if k is even , ∣ V ( G ) ∣ 2 if k is odd . □

Corollary 2.5

A complete graph K n is a k-PG if and only if n ≤ 2 or ( n = 3 and k is odd).

Proof

If n ≤ 2 , then K n is a k-PG, because K 1 and K 2 are paths.

If n = 3 and k is odd, then the integers in the set { k , k + 1 , k + 2 } are mutually relatively prime. So, any bijection map from V ( K 3 ) to { k , k + 1 , k + 2 } is a k-PL. Thus, K 3 is a k-PG.

If n = 3 and k is even, then K 3 is not a k-PG because gcd ( k , k + 2 ) = 2 .

If n ≥ 4 , then

α ( K n ) = 1 < 2 ≤ ∣ V ( K n ) ∣ 2 .

So, using the previous lemma, K n is not a k-PG.□

In the following theorem, we provide a characterization when a graph of order 3 is a k-PG.

Theorem 2.6

A graph of order 3 is a k-PG if and only if it is a non-complete graph or k is odd.

Proof

Follows by Lemma 2.3 and Corollary 2.5.□

Corollary 2.7

Let G 1 be a PG of order m and G 2 be a graph of order 3. If G 2 is a non-complete graph or m is even, then G 1 ∪ G 2 is a P G .

Proof

Suppose m is even or G 2 is a non-complete graph of order 3. Then, m + 1 is odd or G 2 is a non-complete graph of order 3. By Theorem 2.6, we have G 2 is an ( m + 1 ) -PG. Therefore, by Lemma 2.2, we obtain G 1 ∪ G 2 is a PG.□

The previous corollary is a generalization of the following result in [7], and they prove that “if G 1 is a PG of order m where m is even and G 2 is a graph of order 3, then G 1 ∪ G 2 is a PG.”

Definition 2.8

A k-PG is called a maximal k-PG, if adding any edge to G yields a non- k -PG.

Remark 1

A graph is a k-PG if and only if it is a spanning subgraph of a maximal k-PG.

Remark 2

In order to determine the maximal k-PG of order n for a given value of k , we label the vertices of K n with the integers k , k + 1 , … , k + n − 1 . We remove all edges in which the labels on the end-vertices are not relatively prime. The resulting graph is a maximal k -PG of order n .

Definition 2.9

Let G 1 , G 2 , … , G m be disjoint graphs. Then,

  1. The join of G 1 , G 2 , … , G m , denoted by G 1 + G 2 , … + G m , is the graph G 1 ∪ G 2 , … ∪ G m with the addition of an edge from each vertex of G i to each vertex of G j for all i ≠ j .

  2. The pairwise join of G 1 , G 2 , … , G m , denoted by G 1 + . G 2 , … + . G m , is the graph G 1 ∪ G 2 , … ∪ G m with the addition of an edge from each vertex of G i to each vertex of G i + 1 for i = 1 , 2 , … , m − 1 .

Consider the following figure to illustrate the differences between the graphs K 2 + K 1 + K 1 and K 2 + . K 1 + . K 1 (Figure 1).

Figure 1 K 2 + K 1 + K 1 {K}_{2}+{K}_{1}+{K}_{1} and K 2 + . K 1 + . K 1 {K}_{2}\mathop{+}\limits^{.}{K}_{1}\mathop{+}\limits^{.}{K}_{1} .
Figure 1

K 2 + K 1 + K 1 and K 2 + . K 1 + . K 1 .

In the following theorem, we provide a characterization when a graph of order 4 is a maximal k-PG.

Theorem 2.10

A graph of order 4 is a maximal k-PG if and only if it exhibits an isomorphism with one of the following graphs:

  1. K 2 + . K 1 + . K 1 whenever k is divisible by 3.

  2. K 4 − e whenever k is not divisible by 3, where e is an edge of K 4

Proof

Let G be a maximal k-PG of order 4. The labels of the vertices are k , k + 1 , k + 2 , k + 3 . Since the largest difference between these integers is 3, the only possible common prime factors between any pair of these integers are 2 and 3. The integers k and k + 2 are divisible by 2 whenever k is even and the integers k + 1 and k + 3 are divisible by 2 whenever k is odd.

  1. If k is divisible by 3, then k + 3 is also divisible by 3. So, G must have a vertex of degree 1. Thus, G = K 2 + . K 1 + . K 1 .

  2. If k is not divisible by 3, then no pair of these integers is divisible by 3. So, we have only two vertices that are not adjacent in G and hence G = K 4 − e , where e is an edge of K 4 . □

According to Theorem 2.10, it is easy to characterize k -PGs of order 4. We state this in the following corollary.

Corollary 2.11

A graph of order 4 is a k-PG if and only if it exhibits an isomorphism with one of the following graphs:

  1. A subgraph of K 2 + . K 1 + . K 1 .

  2. A non-complete graph whenever k is not divisible by 3.

Remark 3

In the first condition in Corollary 2.11, we need not to say k is divisible by 3, because K 2 + . K 1 + . K 1 is a subgraph of K 4 − e .

Corollary 2.12

Let G 1 be a PG of order m and G 2 be a graph of order 4. If G 2 is a subgraph of K 2 + . K 1 + . K 1 or ( m + 1 is not divisible by 3 and G 2 is a-non complete graph), then G 1 ∪ G 2 is a PG.

Proof

Follows by Corollary 2.11 and Lemma 2.2.□

Remark 4

A graph with no edges on i vertices is called the null graph, denoted by N i .

In Theorem 2.13, we characterize when a graph of order 5 is a maximal k-PG.

Theorem 2.13

A graph of order 5 is a maximal k-PG if and only if it exhibits an isomorphism with one of the following graphs:

  1. K 5 − e whenever k ≡ 1 ( mod 6 ), where e is an edge of K 5 .

  2. K 2 + . K 2 + . K 1 whenever k ≡ 3 or 5 ( mod 6 ) .

  3. K 2 + N 3 whenever k ≡ 4   ( mod 6 ) .

  4. P 3 + . K 1 + . K 1 whenever k ≡ 0 or 2 ( mod 6 ) .

Proof

Let G be a maximal k-PG of order 5. The labels of the vertices are k , k + 1 , … , k + 4 and the only possible common prime factors between any pair of these integers are 2 and 3. In all cases, we will mention when pairs of integers are not relatively prime.

  1. If k ≡ 1 ( mod 6 ), then the integers k + 1 and k + 3 are even and are the only pair of integers that are not relatively prime. By Remark 2, G = K 5 − e , where e is an edge of K 5 .

  2. If k ≡ 3 or 5 ( mod 6 ), then the integers k and k + 3 are not relatively prime whenever k is divisible by 3 and the integers k + 1 and k + 4 are not relatively prime whenever k + 1 is divisible by 3. Also, k + 1 and k + 3 are not relatively prime. By Remark 2, G = K 2 + . K 2 + . K 1 .

  3. If k ≡ 4 ( mod 6 ), then the integers k , k + 2 , k + 4 are mutually not relatively prime. By Remark 2, G = K 2 + N 3 .

  4. If k ≡ 0 or 2 ( mod 6 ), then the integers k and k + 3 are not relatively prime whenever k is divisible by 3 and the integers k + 1 and k + 4 are not relatively prime whenever k + 1 is divisible by 3. Also, the integers k , k + 2 , k + 4 are mutually not relatively prime. By Remark 2, G = P 3 + . K 1 + . K 1 .□

According to Theorem 2.13, we obtain the following corollary.

Corollary 2.14

A graph of order 5 is a k-PG if and only if it exhibits an isomorphism with one of the following graphs:

  1. A subgraph of P 3 + . K 1 + . K 1 .

  2. A subgraph of K 2 + N 3 whenever k is odd or k + 2 is divisible by 3.

  3. A subgraph of K 2 + . K 2 + . K 1 whenever k is odd.

  4. A non-complete graph whenever k ≡ 1 ( mod 6 ) .

Proof

Since K 2 + . K 2 + . K 1 is a non-complete graph, P 3 + . K 1 + . K 1 is a subgraph of K 2 + N 3 , and K 2 + N 3 is a subgraph of K 2 + . K 2 + . K 1 , the proof follows by Theorem 2.13 and Remark 1.□

Corollary 2.15

If G is a k-PG of order 5 whenever k is even, then G is a k-PG of order 5 whenever k is odd.

Corollary 2.16

Let G 1 be a PG of order m and G 2 be a graph of order 5. Then, G 1 ∪ G 2 is a PG whenever G 2 exhibits an isomorphism with one of the following graphs:

  1. A subgraph of P 3 + . K 1 + . K 1 .

  2. A subgraph of K 2 + N 3 whenever m is even or m is divisible by 3.

  3. A subgraph of K 2 + . K 2 + . K 1 whenever m is even.

  4. A non-complete graph whenever m ≡ 0 ( mod 6 ) .

In Theorems 2.17 and 2.18, we characterize when a graph of order 6 is a maximal k-PG.

Theorem 2.17

The only maximal k-PG of order 6 is P 3 + . K 2 + . K 1 whenever k is not divisible by 5.

Proof

Let G be a maximal k-PG of order 6. The labels of the vertices are k , k + 1 , … , k + 5 . Since k is not divisible by 5, the only possible common prime factors between any pair of these integers are 2 and 3. It is clear that three of these integers are divisible by 2 and two of them are divisible by 3. Furthermore, only one of these integers is divisible by 2 and 3 at the same time. Thus, G has exactly two maximal independent sets of cardinality greater than 1. One of them is of cardinality 3 and the other of cardinality 2 and these sets have a common vertex. By Remark 2, G = P 3 + . K 2 + . K 1 . □

In the following theorems and corollaries, let H be the following graph (Figure 2).

Figure 2 H.
Figure 2

H.

Theorem 2.18

If k is divisible by 5, then a graph of order 6 is a maximal k-PG if and only if it exhibits an isomorphism with one of the following graphs:

  1. ( P 3 + K 1 ) + . K 1 + . K 1 whenever k ≡ 0 or 1 ( mod 6 ) .

  2. K 2 + . K 2 + . N 2 whenever k ≡ 3 or 4 ( mod 6 ) .

  3. H whenever k ≡ 2 or 5 ( mod 6 ).

Proof

Let G be a maximal k-PG of order 6. The labels of the vertices are k , k + 1 , … , k + 5 . Since k is divisible by 5, the integers k and k + 5 are not relatively prime. In all cases, we will state all pairs of these integers that are not relatively prime.

  1. Suppose k ≡ 0 or 1 ( mod 6 ). Then, either k , k + 2 , and k + 4 are mutually not relatively prime and k and k + 3 are not relatively prime or k + 1 , k + 3 , and k + 5 are mutually not relatively prime and k + 2 and k + 5 are not relatively prime. By Remark 2, G = ( P 3 + K 1 ) + . K 1 + . K 1 .

  2. Suppose k ≡ 3 or 4 ( mod 6 ). Then, either k , k + 2 , and k + 4 are mutually not relatively prime and k + 2 and k + 5 are not relatively prime) or k + 1 , k + 3 , and k + 5 are mutually not relatively prime and k and k + 3 are not relatively prime). By Remark 2, G = K 2 + . K 2 + . N 2 .

  3. Suppose k ≡ 2 or 5 ( mod 6 ), then k + 1 and k + 4 are not relatively prime. Also, either k , k + 2 , and k + 4 are mutually not relatively prime or k + 1 , k + 3 , and k + 5 are mutually not relatively prime. By Remark 2, G = H .□

According to Theorems 2.17 and 2.18 and since K 2 + . K 2 + . N 2 , ( P 3 + K 1 ) + . K 1 + . K 1 and H are subgraphs of P 3 + . K 2 + . K 1 , it is easy to characterize k -PGs of order 6. We state this in the following corollary.

Corollary 2.19

A graph of order 6 is a k-PG if and only if it exhibits an isomorphism with one of the following graphs:

  1. A subgraph of P 3 + . K 2 + . K 1 whenever k is not divisible by 5.

  2. A subgraph of ( P 3 + K 1 ) + . K 1 + . K 1 whenever k ≡ 0 or 1 ( mod 6 ) .

  3. A subgraph of K 2 + . K 2 + . N 2 whenever k ≡ 3 or 4 ( mod 6 ) .

  4. A subgraph of H whenever k ≡ 2 or 5 ( mod 6 ) .

According to Corollary 2.19, we have the following two corollaries.

Corollary 2.20

Let G be a graph of order 6. Then

  1. If G is a k-PG whenever k is divisible by 5, then G is a k-PG whenever k is not divisible by 5.

  2. If G is a subgraph of K 3 + . K 1 + . N 2 , then G is a k-PG for all k ∈ N .

Proof

(ii) Because K 3 + . K 1 + . N 2 is the maximal common subgraph of K 2 + . K 2 + . N 2 , ( P 3 + K 1 ) + . K 1 + . K 1 and H .□

Corollary 2.21

Let G 1 be a PG of order m and G 2 be a graph of order 6. Then, G 1 ∪ G 2 is a PG whenever G 2 exhibits an isomorphism with one of the following graphs:

  1. A subgraph of P 3 + . K 2 + . K 1 whenever m + 1 is not divisible by 5.

  2. A subgraph of ( P 3 + K 1 ) + . K 1 + . K 1 whenever m ≡ 0 or 5 ( mod 6 ) .

  3. A subgraph of K 2 + . K 2 + . N 2 whenever m ≡ 2 or 3 ( mod 6 ) .

  4. A subgraph of H whenever m ≡ 1 or 4 ( mod 6 ) .

To characterize when a graph of order 7 is a maximal k-PG, we have many cases. In the following theorem, we will study when k and k + 1 are not divisible by 5.

Theorem 2.22

If k and k + 1 are not divisible by 5, then a graph of order 7 is a maximal k-PG if and only if it exhibits an isomorphism with one of the following graphs:

  1. P 3 + . K 2 + . N 2 whenever k ≡ 0 ( mod 6 ) .

  2. K 1 , 3 + . K 2 + K 1 whenever k ≡ 2 or 4 ( mod 6 ) .

  3. C 4 + . K 2 + . K 1 whenever k ≡ 3 ( mod 6 ) .

  4. P 3 + . K 3 + . K 1 whenever k ≡ 1 or 5 ( mod 6 ) .

Proof

Let G be a maximal k-PG of order 7. The labels of the vertices are k , k + 1 , … , k + 6 . In all cases, we will state all pairs of these integers that are not relatively prime.

  1. If k ≡ 0 ( mod 6 ) , then k , k + 2 , k + 4 , and k + 6 are mutually not relatively prime. Also, k , k + 3 , and k + 6 are mutually not relatively prime. By Remark 2, G = P 3 + . K 2 + . N 2 .

  2. If k ≡ 2 or 4 ( mod 6 ) , then k , k + 2 , k + 4 , and k + 6 are mutually not relatively prime. Also, either k + 1 and k + 4 are relatively prime or k + 2 and k + 5 are relatively prime. By Remark 2, G = K 1 , 3 + . K 2 + . K 1 .

  3. If k ≡ 3 ( mod 6 ) , then k + 1 , k + 3 , and k + 5 are mutually not relatively prime. Also k , k + 3 , and k + 6 are mutually not relatively prime. By Remark 2, G = C 4 + . K 2 + . K 1 .

  4. If k ≡ 1 or 5 ( mod 6 ) , then k + 1 , k + 3 , and k + 5 are mutually not relatively prime. Also, either k + 1 and k + 4 are relatively prime or k + 2 and k + 5 are relatively prime. By Remark 2, G = P 3 + . K 3 + . K 1 .□

According to Theorem 2.22 and since the graph P 3 + . K 2 + . N 2 is a subgraph of K 1 , 3 + . K 2 + . K 1 and C 4 + . K 2 + . K 1 which are subgraphs of P 3 + . K 3 + . K 1 , it is easy to characterize k -PGs of order 7 whenever k and k + 1 are not divisible by 5. This is articulated in the following corollary.

Corollary 2.23

If k and k + 1 are not divisible by 5, then a graph of order 7 is a k -PG if and only if it exhibits an isomorphism with one of the following graphs:

  1. A subgraph of P 3 + . K 2 + . N 2 .

  2. A subgraph of K 1 , 3 + . K 2 + . K 1 whenever k is not divisible by 3.

  3. A subgraph of C 4 + . K 2 + . K 1 whenever k is odd.

  4. A subgraph of P 3 + . K 3 + . K 1 whenever k is odd and k is not divisible by 3.

Corollary 2.24

Let G 1 be a PG of order m and G 2 be a graph of order 7 where m + 1 and m + 2 are not divisible by 5. Then, G 1 ∪ G 2 is a PG whenever G 2 exhibits an isomorphism with one of the following graphs:

  1. A subgraph of P 3 + . K 2 + . N 2 .

  2. A subgraph of K 1 , 3 + . K 2 + . K 1 whenever m + 1 is not divisible by 3.

  3. A subgraph of C 4 + . K 2 + . K 1 whenever m is even.

  4. A subgraph of P 3 + . K 3 + . K 1 whenever m is even and m + 1 is not divisible by 3.

To study when a graph of order 7 is a maximal k-PG whenever k is divisible by 5 or whenever k + 1 is divisible by 5, we need the six graphs in the following figures (Figures 3, 4, 5, 6, 7, 8).

Figure 3 J 1 {J}_{1} .
Figure 3

J 1 .

Figure 4 J 2 {J}_{2} .
Figure 4

J 2 .

Figure 5 J 3 {J}_{3} .
Figure 5

J 3 .

Figure 6 J 4 {J}_{4} .
Figure 6

J 4 .

Figure 7 J 5 {J}_{5} .
Figure 7

J 5 .

Figure 8 J 6 {J}_{6} .
Figure 8

J 6 .

Theorem 2.25

If k is divisible by 5, then a graph of order 7 is a maximal k-PG if and only if it exhibits an isomorphism with one of the following graphs:

  1. J 1 whenever k ≡ 0 ( mod 6 ) .

  2. J 2 whenever k ≡ 1 ( mod 6 ) .

  3. J 3 whenever k ≡ 2 ( mod 6 ) .

  4. J 4 whenever k ≡ 3 ( mod 6 ) .

  5. J 5 whenever k ≡ 4 ( mod 6 ) .

  6. J 6 whenever k ≡ 5 ( mod 6 ) .

Theorem 2.26

If k + 1 is divisible by 5, then a graph of order 7 is a maximal k-PG if and only if it exhibits an isomorphism with one of the following graphs:

  1. J 1 whenever k ≡ 0 ( mod 6 ) .

  2. J 6 whenever k ≡ 1 ( mod 6 ) .

  3. J 5 whenever k ≡ 2 ( mod 6 ) .

  4. J 4 whenever k ≡ 3 ( mod 6 ) .

  5. J 3 whenever k ≡ 4 ( mod 6 ) .

  6. J 2 whenever k ≡ 5 ( mod 6 ) .

Remark 5

J 1 is a subgraph of J 3 and J 5 , J 5 is a subgraph of J 4 , J 3 , and J 4 are subgraphs of J 2 and J 6 .

According to Theorems 2.25 and 2.26 and Remark 5, we have the following corollaries:

Corollary 2.27

If k is divisible by 5, then a graph of order 7 is a k-PG if and only if it exhibits an isomorphism with one of the following graphs:

  1. A subgraph of J 1 .

  2. A subgraph of J 3 whenever k ≡ 1 , 2 or 5 ( mod 6 )

  3. A subgraph of J 5 whenever k is odd or k ≡ 4 ( mod 6 ) .

  4. A subgraph of J 4 whenever k is odd.

  5. A subgraph of J 2 whenever k ≡ 1 ( mod 6 ) .

  6. A subgraph of J 6 whenever k ≡ 5 ( mod 6 ) .

Corollary 2.28

If k + 1 is divisible by 5, then a graph of order 7 is a k-PG if and only if it exhibits an isomorphism with one of the following graphs:

  1. A subgraph of J 1 .

  2. A subgraph of J 3 whenever k ≡ 1 , 4 or 5 ( mod 6 )

  3. A subgraph of J 5 whenever k is odd or k ≡ 2 ( mod 6 ) .

  4. A subgraph of J 4 whenever k is odd.

  5. A subgraph of J 2 whenever k ≡ 5 ( mod 6 ) .

  6. A subgraph of J 6 whenever k ≡ 1 ( mod 6 ) .

Corollary 2.29

Let G 1 be a PG of order m and G 2 be a graph of order 7 where m + 1 is divisible by 5. Then G 1 ∪ G 2 is a PG whenever G 2 exhibits an isomorphism with one of the following graphs:

  1. A subgraph of J 1 .

  2. A subgraph of J 3 whenever m ≡ 0 , 1 or 4 ( mod 6 )

  3. A subgraph of J 5 whenever m is even or m ≡ 3 ( mod 6 ) .

  4. A subgraph of J 4 whenever m is even.

  5. A subgraph of J 2 whenever m is divisible by 6.

  6. A subgraph of J 6 whenever m ≡ 4 ( mod 6 ) .

Corollary 2.30

Let G 1 be a PG of order m and G 2 be a graph of order 7 where m + 2 is divisible by 5. Then, G 1 ∪ G 2 is a PG whenever G 2 exhibits an isomorphism with one of the following graphs:

  1. A subgraph of J 1 .

  2. A subgraph of J 3 whenever m ≡ 0 , 3 or 4 ( mod 6 ) .

  3. A subgraph of J 5 whenever m is even or m ≡ 1 ( mod 6 ) .

  4. A subgraph of J 4 whenever m is even.

  5. A subgraph of J 2 whenever m ≡ 4 ( mod 6 ) .

  6. A subgraph of J 6 whenever m is divisible by 6.

Finally, we will study when a cycle is a k-PG. In Theorem 2.31, we prove that an odd cycle is not a k-PG whenever k is even.

Theorem 2.31

An odd cycle C 2 m + 1 is not a k-PG whenever k is even.

Proof

Let G be any k-PG of order 2 m + 1 whenever k is even. Then, by Lemma 2.4

α ( G ) ≥ ∣ V ( G ) ∣ 2 = 2 m + 1 2 = m + 1 > m = α ( C 2 m + 1 ) .

Thus, C 2 m + 1 is not a k - P G . □

Now, we will present some cases in which C m is a k-PG.

Theorem 2.32

A cycle C m is a k-PG whenever gcd ( k , k + m − 1 ) = 1 .

Proof

Let C m be the following cycle v 1 v 2 … v m v 1 . Label the vertices of C m successively with the integers k , k + 1 , … , k + m − 1 . Since gcd ( k , k + m − 1 ) = 1 and successive integers are relatively prime, we obtain C m is a k-PG.□

In [10], the authors proved that “an odd cycle C 2 m + 1 is a k-PG whenever k is odd and m = 1 , 2 , 4 .” In Theorem 2.33, we generalized their result.

Theorem 2.33

A cycle C 2 m + 1 is a k-PG whenever k is odd.

Proof

Since k is odd,

gcd ( k , k + ( 2 m + 1 ) − 1 ) = gcd ( k , k + 2 m ) = 1 .

So, by Theorem 2.32  C 2 m + 1 is a k-PG.□

Also, in [10], the authors said “an even cycle C 2 m is a k-PG for all m ≥ 2 .” Nevertheless, it is crucial to emphasize that this result does not hold in general, for example, C 4 is not 3 - P G . In the following theorem, we add a condition on k to make the previous result correct.

Theorem 2.34

An even cycle C 2 m is a k-PG whenever g c d ( k , 2 m − 1 ) = 1 .

Proof

Let d = gcd ( k , k + 2 m − 1 ) . Then, d divides k and d divides

( k + 2 m − 1 ) − k = 2 m − 1 .

But k and 2 m − 1 are relatively prime and hence d = 1 . Use Theorem 2.32 to obtain C 2 m is a k-PG.□

Acknowledgement

We sincerely thank the reviewers for their precise evaluation and insightful comments on the manuscript. Their valuable input is gratefully acknowledged.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: All authors listed have significantly contributed to the development and the writing of this article.

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

  4. Data availability statement: No data were used for the research described in the article.

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Received: 2024-04-21
Revised: 2024-10-28
Accepted: 2024-11-08
Published Online: 2024-12-06

© 2024 the author(s), published by De Gruyter

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

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
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  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
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  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
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  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
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  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
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  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
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  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
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  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0097
Keywords for this article
independence number; prime labeling; prime graphs; k-prime graphs; maximal k-prime graphs
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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