On the δ-chromatic numbers of the Cartesian products of graphs

Wipawee Tangjai; Witsarut Pho-on; Panupong Vichitkunakorn

doi:10.1515/math-2024-0087

Artikel Open Access

On the δ-chromatic numbers of the Cartesian products of graphs

Wipawee Tangjai , Witsarut Pho-on und Panupong Vichitkunakorn

Veröffentlicht/Copyright: 18. November 2024

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Abstract

In this work, we study the δ -chromatic number of a graph, which is the chromatic number of the δ -complement of a graph. We give a structure of the δ -complements and sharp bounds on the δ -chromatic numbers of the Cartesian products of graphs. Furthermore, we compute the δ -chromatic numbers of various classes of Cartesian product graphs, including the Cartesian products among cycles, paths, and stars.

Keywords: δ-complement graph; chromatic number; Cartesian product; coloring

MSC 2010: 05C07; 05C15; 05C35; 05C38; 05C69

1 Introduction

The concept of δ -complement was introduced in 2022 [1]. Their research focused on exploring various intriguing characteristics of these graphs, including properties like δ -self-complementary, adjacency, and hamiltonicity. In 2023, Vichitkunakorn et al. [2] introduced the term δ -chromatic number of a graph G which refers to the chromatic number of the δ -complement of G . They established a Nordhaus-Gaddum bound-type relation between the chromatic number and the δ -chromatic number across various parameters: the clique number, the number of vertices, and the degrees of vertices. The given bounds are sharp and the classes of graphs satisfying those bounds are given [2]. In this study, we present a more detailed outcome concerning the δ -chromatic number of the Cartesian product of graphs.

In 1957, Sabidussi [3] showed that the chromatic number of the Cartesian product graphs is equal to the maximum chromatic number between such two graphs. A lot of subsequent research has been exploring different types of chromatic numbers of the Cartesian product graphs such as list chromatic number [4], packing chromatic number [5], and b -chromatic number [6,7].

We first recall some basic notations and definitions needed in this article. Let G be a graph. For a subset U of V ( G ) , G [ U ] denotes the subgraph induced by U . A vertex coloring c of G is a proper coloring if each pair of adjacent vertices has distinct colors. The chromatic number of G , denoted by χ ( G ) , is the minimum number of colors needed so that ( G , c ) is properly colored. For each vertex u ∈ V ( G ) , we use notation d G ( u ) for the degree of u in G . Throughout this article, we let P n be a path with n vertices, K n be a complete graph with n vertices, and C n be a cycle with n vertices. We let S 1 , n be a star with n pendants. For graphs G and H , the Cartesian product of G and H , denoted by G □ H , is a graph where V ( G □ H ) = V ( G ) × V ( H ) and u v ∈ E ( G □ H ) if either x = x ′ and y y ′ ∈ E ( H ) or y = y ′ and x x ′ ∈ E ( G ) for u = ( x , y ) and v = ( x ′ , y ′ ) .

In this work, we give a structure of the δ -complement of the finite Cartesian products of graphs. Sharp bounds on the δ -chromatic number (the chromatic number of δ -complement) of the finite Cartesian products of graphs are also given. In addition, we determine the specific value of the δ -chromatic numbers of various classes of the Cartesian product of well-known graphs such as cycles, paths, and stars.

2 Preliminary results

In this section, we provide the definitions of the δ -complement of a graph and its δ -chromatic number. We also include previous results on the chromatic number of the Cartesian product of graphs and Nordhaus-Gaddum bounds on δ -chromatic numbers.

Definition 1

[1] The δ -complement of a graph G , denoted G δ , is a graph obtained from G by using the same vertex set and the following edge conditions: u v ∈ E ( G δ ) if

d ( u ) = d ( v ) in G and u v ∉ E ( G ) , or
d ( u ) ≠ d ( v ) in G and u v ∈ E ( G ) .

The δ -complement graph is a variation of a complement graph in which the complement operation is applied only to vertices of the same degree. In the δ -complement graph, the adjacency of the vertices with distinct degrees remains the same, while the adjacency of the vertices with the same degree is changed. An example of a graph and its δ -complement is shown in Figure 1.

$Figure 1 Graph G G and its δ \delta -complement G δ {G}_{\delta } .$

Figure 1

Graph G and its δ -complement G δ .

Definition 2

[2] A δ -chromatic number χ δ ( G ) of a graph G is the chromatic number of G δ .

Results on the δ -chromatic numbers of some important graphs are χ δ ( P n ) = n − 2 2 for n ≥ 5 [2], χ δ ( C n ) = n 2 [2], and χ δ ( W n ) = 1 + χ δ ( C n ) = 1 + n 2 . The following theorem is well-known and will be used in the proof of Theorem 11.

Theorem 3

[3] Let G and H be graphs. We have χ ( G □ H ) = max { χ ( G ) , χ ( H ) } .

Some Nordhaus-Gaddum bounds on δ -chromatic numbers were shown by Vichitkunakorn et al. [2] and are presented below.

Theorem 4

[2] For n ≥ 4 , let G be a graph with n vertices. Let d 1 , … , d m be all distinct values of the degrees of the vertices in G. Partition V ( G ) into non-empty sets V d 1 , V d 2 , … , V d m . We have

max 1 ≤ i ≤ m { ∣ V d i ∣ } ≤ χ ( G ) ⋅ χ δ ( G ) ≤ m + n 2 2

and

2 ⋅ max 1 ≤ i ≤ m { ∣ V d i ∣ } ≤ χ ( G ) + χ δ ( G ) ≤ m + n .

3 Structure of the δ -complements of Cartesian products

This section contains the structure of the δ -complement of the Cartesian product of graphs.

The following theorem shows that the edge set of the δ -complements of the Cartesian product contains the edge set of the Cartesian product of the δ -complements of graphs. It is a fundamental result that will be used throughout what follows.

Theorem 5

For graphs G and H, we have ( G □ H ) δ = ( V , E ) where V = V ( G □ H ) and E = E ( G δ □ H δ ) ∪ S where S = { u v : u = ( u 1 , u 2 ) ∈ V ( G □ H ) and v = ( v 1 , v 2 ) ∈ V ( G □ H ) where u 1 ≠ v 1 , u 2 ≠ v 2 and d G □ H ( u ) = d G □ H ( v ) } .

Proof

We want to show that E = E ( G δ □ H δ ) ∪ S .

( ⊆ ) Let u = ( u 1 , u 2 ) and v = ( v 1 , v 2 ) be distinct vertices in ( G □ H ) δ , where u v ∈ E ( ( G □ H ) δ ) . It follows that either u v ∈ E ( G □ H ) and d G □ H ( u ) ≠ d G □ H ( v ) or u v ∉ E ( G □ H ) and d G □ H ( u ) = d G □ H ( v ) .

In case u v ∈ E ( G □ H ) and d G □ H ( u ) ≠ d G □ H ( v ) , without loss of generality, we suppose that u 1 = v 1 , u 2 ≠ v 2 and u 2 v 2 ∈ E ( H ) . Since d G ( u 1 ) = d G ( v 1 ) , it follows that d H ( u 2 ) ≠ d H ( v 2 ) . Thus, u 2 v 2 ∈ E ( H δ ) . Hence, u v ∈ E ( G δ □ H δ ) .

In case u v ∉ E ( G □ H ) and d G □ H ( u ) = d G □ H ( v ) , if u 1 ≠ v 1 and u 2 ≠ v 2 , then it is clear that u v ∈ S . Next, we suppose u 1 = v 1 or u 2 = v 2 . Without loss of generality, we assume that u 1 = v 1 . Then, u 2 v 2 ∉ E ( H ) . Since d G □ H ( u ) = d G □ H ( v ) , we then obtain d H ( u 2 ) = d H ( v 2 ) . Thus, u 2 v 2 ∈ E ( H δ ) . Hence, u v ∈ E ( G δ □ H δ ) .

( ⊇ ) Let u v ∈ E ( G δ □ H δ ) ∪ S . Consider u v ∈ E ( G δ □ H δ ) . Without loss of generality, we suppose that u 1 = v 1 , u 2 ≠ v 2 , and u 2 v 2 ∈ E ( H δ ) . If d G □ H ( u ) = d G □ H ( v ) , then d H ( u 2 ) = d H ( v 2 ) . Thus, u 2 v 2 ∉ E ( H ) . Hence, u v ∉ E ( G □ H ) . Since d G □ H ( u ) = d G □ H ( v ) , we have u v ∈ E ( ( G □ H ) δ ) . If d G □ H ( u ) ≠ d G □ H ( v ) , then d H ( u 2 ) ≠ d H ( v 2 ) . Thus, u 2 v 2 ∈ E ( H ) and u v ∈ E ( G □ H ) . Since d G □ H ( u ) ≠ d G □ H ( v ) , we have u v ∈ E ( ( G □ H ) δ ) . Now, we consider u v ∈ S . We have that u 1 ≠ v 1 and u 2 ≠ v 2 . So, u v ∉ E ( G □ H ) . Since d G □ H ( u ) = d G □ H ( v ) , it follows that u v ∈ E ( ( G □ H ) δ ) .□

Corollary 6

Let G and H be graphs. We have ( G □ H ) δ = G δ □ H δ if and only if for any u = ( u 1 , u 2 ) and v = ( v 1 , v 2 ) in V ( G □ H ) , where u 1 ≠ v 1 and u 2 ≠ v 2 , we have d G □ H ( u ) ≠ d G □ H ( v ) .

The following example illustrates the structure described in Theorem 5.

Example 7

Consider G = P 3 and H = P 4 . Figure 2 displays the edges of ( G □ H ) δ as E ( G δ □ H δ ) ∪ S .

$Figure 2 Left: Graph G □ H G\square H where G = P 3 G={P}_{3} and H = P 4 H={P}_{4} . The figures of G G and H H are shown on the left and the top of G □ H G\square H to help construct the graph. Right: Graph ( G □ H ) δ {\left(G\square H)}_{\delta } . The edges of ( G □ H ) δ {\left(G\square H)}_{\delta } that are in E ( G δ □ H δ ) E\left({G}_{\delta }\square {H}_{\delta }) and S S are shown in blue and red, respectively, where S S is an edge set defined in Theorem 5.$

Figure 2

Left: Graph G □ H where G = P 3 and H = P 4 . The figures of G and H are shown on the left and the top of G □ H to help construct the graph. Right: Graph ( G □ H ) δ . The edges of ( G □ H ) δ that are in E ( G δ □ H δ ) and S are shown in blue and red, respectively, where S is an edge set defined in Theorem 5.

In general, we have the following theorem for a finite Cartesian product of graphs.

Theorem 8

For graphs G 1 , … , G k , we have ( G 1 □ ⋯ □ G k ) δ = ( V , E ) where V = V ( G 1 □ ⋯ □ G k ) and E = E ( ( G 1 ) δ □ ⋯ □ ( G k ) δ ) ∪ S such that S = { u v : u = ( u 1 , … , u k ) ∈ V , v = ( v 1 , … , v k ) ∈ V where there are at least two indices i, j that u i ≠ v i , u j ≠ v j , and d G 1 □ ⋯ □ G k ( u ) = d G 1 □ ⋯ □ G k ( v ) } .

Proof

It is well-known that two vertices u = ( u 1 , … , u k ) and ( v 1 , … , v k ) in G 1 □ ⋯ □ G k are adjacent if and only if there is exactly one i such that u i ≠ v i and u i v i ∈ E ( G i ) . The rest of the proof follows similar arguments as in Theorem 5.□

The following two results are applications of Theorem 8.

Theorem 9

( G 1 □ ⋯ □ G k ) δ = ( G 1 ) δ □ ⋯ □ ( G k ) δ if and only if there are at most one i such that G i ≠ K 1 .

Proof

From Theorem 8, we need to show that S = ∅ if and only if there are at most one i such that G i ≠ K 1 .

Suppose that there are i ≠ j such that G i ≠ K 1 and G j ≠ K 1 . Choose u = ( u 1 , … , u k ) and v = ( v 1 , … , v 2 ) such that u i ≠ v i , u j ≠ v j , d G i ( u i ) = d G i ( v i ) , d G j ( u j ) = d G j ( v j ) and u ℓ = v ℓ for all ℓ ∉ { i , j } . So, d G 1 □ ⋯ □ G k ( u ) = d G 1 □ ⋯ □ G k ( v ) . Then, u v ∈ S . Hence, S ≠ ∅ .

The converse is obvious.□

Corollary 10

( G □ H ) δ = G δ □ H δ if and only if G = K 1 or H = K 1 .

4 Bounds on the δ -chromatic numbers of Cartesian products

This section contains our results on lower bounds and upper bounds of δ -chromatic numbers for the Cartesian product of graphs. We also provide examples where the upper bounds are sharp.

Theorem 11

Let G 1 , … , G k be graphs. We have

max { χ δ ( G 1 ) , … , χ δ ( G k ) } ≤ χ δ ( G 1 □ ⋯ □ G k ) .

Proof

The proof follows directly from Theorems 3 and 8.□

The next result gives an upper bound on a Cartesian product of a class of graphs. This result will be used throughout the remaining of this article. The range of the values for the modulo p used in the next theorem is 1 to p .

Theorem 12

Let G and H be graphs. If any positive degree difference of vertices in G is not equal to that of in H, then

χ δ ( G □ H ) ≤ n max ( H ) ⋅ max ( χ δ ( G ) , m ( H ) ) ,

where n max ( H ) denotes the maximum number of vertices of the same degree in H and m ( H ) is the number of different degrees in H. Furthermore, the bound is sharp.

Proof

By Theorem 5 and the assumption that any positive degree difference of vertices in G is not equal to that of in H , the edges in S are u v where u = ( u 1 , u 2 ) , v = ( v 1 , v 2 ) such that u 1 ≠ v 1 , u 2 ≠ v 2 , d G ( u 1 ) = d G ( v 1 ) and d H ( u 2 ) = d H ( v 2 ) . We partition V ( H ) according to vertex degree into W 1 , W 2 , … , W m ( H ) . Write W j = { h j , 1 , h j , 2 , … , h j , n j } for 1 ≤ j ≤ m ( H ) .

Define p = max ( χ δ ( G ) , m ( H ) ) . Let c 0 : V ( G ) → { 1 , 2 , … , χ δ ( G ) } be a proper coloring of G δ . We define a coloring c : V ( G ) × V ( H ) → { 1 , 2 , … , n max ( H ) ⋅ p } as

c ( g , h j , k ) = f ( g , j ) + ( k − 1 ) p ,

for k = 1 , … , n j , where f ( g , j ) ∈ { 1 , 2 , … , p } and f ( g , j ) ≡ ( c 0 ( g ) + j − 1 ) ( mod p ) . The first copy of G in W 1 obtains the original coloring c 0 , while we keep adding p to the coloring of each other copy of G in W 1 . In other W j , we perform different cyclic permutations modulo p to c 0 and assign it to the first copy of G in W j . See Table 1 for an example. We see that the vertices in the same copy of G received a coloring equivalent to c 0 and a cyclic permutation modulo p up to an additive constant ( k − 1 ) p for some k = 1 , … , n j . For a fixed g ∈ V ( G ) , the vertices in the same copy of H , written in the form ( g , h j , k ) where 1 ≤ j ≤ m ( H ) and 1 ≤ k ≤ n j , received distinct colors because j ≤ p and k ≤ n max ( H ) .

Finally, any endpoints of an edge in S are of the form ( g , h j , k ) and ( g ′ , h j , k ′ ) where g ≠ g ′ and k ≠ k ′ , which received different colors as k ≠ k ′ . The sharpness of the bound appears in Theorem 14.□

Table 1

An example of a coloring in the proof of Theorem 12 where χ δ ( G ) = 3 , m ( H ) = 4 , and n max ( H ) = 2

	h 1 , 1	h 1 , 2	h 2 , 1	h 3 , 1	h 3 , 2	h 4 , 1	h 4 , 2
g 1	1	5	2	3	7	4	8
g 2	3	7	4	1	5	2	6
g 3	1	5	2	3	7	4	8
g 4	2	6	3	4	8	1	5
g 5	3	7	4	1	5	2	6

Corollary 13

Let G be a graph with χ δ ( G ) ≥ 2 . If d G ( v ) ≠ d G ( u ) + 1 for all u , v ∈ V ( G ) , then χ δ ( G □ P 3 ) ≤ 2 χ δ ( G ) .

The following theorem implies that the bound given in Theorem 12 is sharp.

Theorem 14

For n ≥ 5 , we have χ δ ( C n □ P 3 ) = 2 χ δ ( C n ) = 2 n 2 .

Proof

Let P 3 = v 1 v 2 v 3 . It is easy to see that χ δ ( C n ) = χ ( C n ¯ ) = n 2 . Since C n is regular, it also follows that d C n □ P 3 ( u , v 1 ) = d C n □ P 3 ( w , v 3 ) for all u , w ∈ V ( C n ) . Since ( u , v 1 ) is not adjacent to ( w , v 3 ) in C n □ P 3 , it follows that each vertex in the first copy of C n is adjacent to all the vertices in the third copy of C n in ( C n □ P 3 ) δ . Hence, the colors used in the two copies do not coincide. Thus, χ δ ( C n □ P 3 ) ≥ 2 χ δ ( C n ) . By Corollary 13, we can conclude that χ δ ( C n □ P 3 ) = 2 χ δ ( C n ) .□

As an example, we apply Theorem 12 to the Cartesian product of a star and a path.

Example 15

For m ≥ 3 , we have χ δ ( S 1 , m □ P 3 ) ≤ 2 ( m + 1 ) .

Proof

Let G = S 1 , m and H = P 3 . Theorem 12 gives the desired upper bound.□

The bound in Example 15 is not sharp. When G = K 1 ∨ H is a join of a singleton and a regular graph H , the following theorem gives an improved upper bound on χ δ ( G □ P 3 ) in terms of χ δ ( H ) . Examples of the graph G include stars S 1 , m = K 1 ∨ N m (in Theorem 18), wheels W m = K 1 ∨ C m , and windmills K 1 ∨ m K n .

Theorem 16

Let H be a k-regular graph. Let G = { u } ∨ H be the join of a singleton and H. Suppose ∣ V ( H ) ∣ ≥ 3 and χ δ ( H ) ≥ 2 . If ∣ V ( H ) ∣ > k + 2 , then χ δ ( G □ P 3 ) ≤ 2 χ δ ( H ) .

Proof

Let r ∈ V ( H ) . Note that d G ( u ) = ∣ V ( H ) ∣ . Since d G □ P 3 ( r , v i ) = d G ( r ) + d P 3 ( v i ) ≤ k + 3 < d G ( u ) + 1 ≤ d G □ P 3 ( u , v j ) for i , j ∈ { 1 , 2 , 3 } , we have ( r , v i ) and ( u , v j ) are adjacent in ( G □ P 3 ) δ if and only if i = j .

Let H δ i be the i th copy of H δ in G δ for i = 1 , 2 , 3 . Next, we construct a proper coloring c as follows. We trivially color H δ 1 , as a copy of H δ , by a χ δ ( H ) -coloring. Since each vertex in H δ 1 is adjacent to all vertices in H δ 3 , it requires 2 χ δ ( H ) colors for H δ 1 and H δ 3 . We color H δ 3 using c ( r , v 3 ) = c ( r , v 1 ) + χ δ ( H ) . For i = 1 , 3 , we note that a vertex ( r , v i ) ∈ V ( H δ i ) and ( s , v 2 ) ∈ V ( H δ 2 ) are adjacent if and only if r = s . We let c ( r , v 2 ) = c ( r , v 1 ) + 1 if 1 ≤ c ( r , v 1 ) ≤ χ δ ( H ) − 1 ; otherwise, c ( r , v 2 ) = 1 . Finally, we color ( u , v 1 ) , ( u , v 2 ) , and ( u , v 3 ) by χ δ ( H ) + 1 , χ δ ( H ) + 2 , and 1, respectively. This gives a proper coloring of ( G □ P 3 ) δ with 2 χ δ ( H ) colors.□

The sharpness of the bound in Theorem 16 will be shown in Theorem 18.

5 δ -chromatic numbers of the Cartesian products of some graphs

In this section, we give our results on the exact values of the δ -chromatic numbers of the Cartesian products of stars and paths.

Theorem 17

χ δ ( S 1 , m □ S 1 , n ) = m n for m , n ≥ 3 .

Proof

Let V ( S 1 , n ) = { 0 , 1 , … , k } where d S 1 , k ( 0 ) = k for k = n , m . An m n -coloring on ( S 1 , m □ S 1 , n ) δ is

c ( i , j ) = j + 1 if i = 0 , ( i − 1 ) n + j if 1 ≤ i ≤ m and 1 ≤ j ≤ n , ( i + 1 ) n if 1 ≤ i < m and j = 0 , n + 2 if i = m and j = 0 ,

as shown in Figure 3. In addition, the set { ( i , j ) : 1 ≤ i ≤ m , 1 ≤ j ≤ n } forms an m n -clique in ( S 1 , m □ S 1 , n ) δ .□

$Figure 3 A proper m n mn -coloring of ( S 1 , m □ S 1 , n ) δ {\left({S}_{1,m}\square {S}_{1,n})}_{\delta } where the vertex labels indicate the colors. The vertices of the same degree in S 1 , m □ S 1 , n {S}_{1,m}\square {S}_{1,n} are indicated by the same color and are pairwise adjacent in ( S 1 , m □ S 1 , n ) δ {\left({S}_{1,m}\square {S}_{1,n})}_{\delta } . Each double line denotes the edges connecting the corresponding vertices between two copies of S 1 , n {S}_{1,n} . Note that the blue and the green will have the same degree when m = n m=n .$

Figure 3

A proper m n -coloring of ( S 1 , m □ S 1 , n ) δ where the vertex labels indicate the colors. The vertices of the same degree in S 1 , m □ S 1 , n are indicated by the same color and are pairwise adjacent in ( S 1 , m □ S 1 , n ) δ . Each double line denotes the edges connecting the corresponding vertices between two copies of S 1 , n . Note that the blue and the green will have the same degree when m = n .

Theorem 18

χ δ ( S 1 , m □ P n ) = m n − 2 2 for m ≥ 3 and n ≥ 3 .

Proof

Let V ( S 1 , m ) = { 0 , 1 , … , m } where d S 1 , m ( 0 ) = m and let V ( P n ) = { 1 , 2 , … , n } where d P n ( 1 ) = d P n ( n ) = 1 .

When n = 3 , Theorem 16 gives χ δ ( S 1 , m □ P 3 ) ≤ 2 m . Since the set { ( i , j ) ∈ V ( ( S 1 , m □ P 3 ) δ ) : 1 ≤ i ≤ m , j = 1 , 3 } forms a 2 m -clique in ( S 1 , m □ P 3 ) δ , we obtain χ δ ( S 1 , m □ P 3 ) = 2 m .

When n = 4 , Theorem 12 with G = P 4 and H = S 1 , m gives χ δ ( S 1 , m □ P 4 ) = χ δ ( P 4 □ S 1 , m ) ≤ 2 m . The set { ( i , j ) ∈ V ( ( S 1 , m □ P 4 ) δ ) : 1 ≤ i ≤ m , j = 1 , 4 } forms a 2 m -clique in ( S 1 , m □ P 4 ) δ . Hence χ δ ( S 1 , m □ P 4 ) = 2 m .

When n ≥ 5 , we let k = n − 2 2 . A coloring is

c ( i , j ) = i + ( k − 1 ) m 0 ≤ i ≤ m and j = 1 , i 1 ≤ i ≤ m and j = n , i + ( ⌊ j ⁄ 2 ⌋ − 1 ) m 0 ≤ i ≤ m and 2 ≤ j ≤ n − 1 where ( i , j ) ≠ ( 0 , 2 ) , ( 0 , 3 ) , k m i = 0 and j = 2 , 3 , n ,

as shown in Figure 4. We thus have a proper k m -coloring of ( S 1 , m □ P n ) δ . In addition, the set { ( i , j ) ∈ V ( ( S 1 , m □ P n ) δ ) : 1 ≤ i ≤ m and 2 ≤ j ≤ n − 1 and j is even } forms a clique of size m n − 2 2 in ( S 1 , m □ P n ) δ .□

$Figure 4 A proper k m km -coloring of ( S 1 , m □ P n ) δ {\left({S}_{1,m}\square {P}_{n})}_{\delta } where k = n ‒ 2 2 k=⌈\frac{n‒2}{2}⌉ where the vertex labels indicate the colors. The vertices of the same degree in S 1 , m □ P n {S}_{1,m}\square {P}_{n} are indicated by the same color and are pairwise adjacent except for the pairs with a dashed line.$

Figure 4

A proper k m -coloring of ( S 1 , m □ P n ) δ where k = n ‒ 2 2 where the vertex labels indicate the colors. The vertices of the same degree in S 1 , m □ P n are indicated by the same color and are pairwise adjacent except for the pairs with a dashed line.

The following lemma is crucial for proving Theorem 20.

Lemma 19

For n ≥ 6 and k ≥ 8 , we have

2 n − 2 2 + 2 k − 2 2 + 1 < ( n − 2 ) ( k − 2 ) 2 .

Proof

Suppose 2 n − 2 2 + 2 k − 2 2 + 1 ≥ ( n − 2 ) ( k − 2 ) 2 .

Case 1. n and k are even. We have n + k − 3 ≥ ( n − 2 ) ( k − 2 ) 2 . So , 2 n + 2 k − 6 ≥ n k − 2 n − 2 k + 4 . Thus, n ≤ 4 k − 10 k − 4 < 6 when k ≥ 8 , which is a contradiction.

Case 2. n is odd and k is even. We have n + k − 2 ≥ ( n − 2 ) ( k − 2 ) 2 . So , 2 n + 2 k − 4 ≥ n k − 2 n − 2 k + 4 . Thus, n ≤ 4 k − 8 k − 4 = 4 + 8 k − 4 . If k > 8 , then n < 6 , which contradicts to the assumption n ≥ 6 . If k = 8 , then n = 6 . This is also a contradiction as n must be odd.

Case 3. n is even and k is odd.

The proof of this case is similar to Case 2.

Case 4. n and k are odd. We have n + k − 1 ≥ ( n − 2 ) ( k − 2 ) + 1 2 . So , 2 n + 2 k − 2 ≥ n k − 2 n − 2 k + 5 . Thus, n ≤ 4 k − 7 k − 4 = 4 + 9 k − 4 . Since k is an odd number where k ≥ 9 , we have n < 6 , which is a contradiction.

Therefore, 2 n − 2 2 + 2 k − 2 2 + 1 < ( n − 2 ) ( k − 2 ) 2 . □

Theorem 20

For 6 ≤ n ≤ k , we have

χ δ ( P n □ P k ) = ( n − 2 ) ( k − 2 ) 2 .

Proof

Let V d be the set of vertices of degree d in P n □ P k . The vertex set of P n □ P k can be partitioned into V 2 , V 3 and V 4 . We note that V ( P n □ P k ) = V ( ( P n □ P k ) δ ) = V ( P n ) × V ( P k ) . Let ( i , j ) ∈ V ( P n □ P k ) for i = 1 , … , n and j = 1 , … , k . We have that V 3 = { ( i , j ) : i = 1 , n and 2 ≤ j ≤ k − 1 } ∪ { ( i , j ) : j = 1 , k and 2 ≤ i ≤ n − 1 } and V 4 = { ( i , j ) : 2 ≤ i ≤ n − 1 and 2 ≤ j ≤ k − 1 } . Thus, ∣ V 2 ∣ = 4 , ∣ V 3 ∣ = 2 ( n + k − 4 ) and ∣ V 4 ∣ = ( n − 2 ) ( k − 2 ) . The vertices ( i , j ) and ( i ′ , j ′ ) are adjacent in P n □ P k if and only if ∣ i − i ′ ∣ + ∣ j − j ′ ∣ = 1 . Thus,

if d P n □ P k ( i , j ) = d P n □ P k ( i ′ , j ′ ) , then the vertices ( i , j ) and ( i ′ , j ′ ) are adjacent in ( P n □ P k ) δ if and only if ∣ i − i ′ ∣ + ∣ j − j ′ ∣ ≥ 2 ,
if d P n □ P k ( i , j ) ≠ d P n □ P k ( i ′ , j ′ ) , then the vertices ( i , j ) and ( i ′ , j ′ ) are adjacent in ( P n □ P k ) δ if and only if ∣ i − i ′ ∣ + ∣ j − j ′ ∣ = 1 .

Since each pair of vertices ( i , j ) , ( i ′ , j ′ ) ∈ V 4 with ∣ i − i ′ ∣ + ∣ j − j ′ ∣ ≥ 2 are adjacent, it follows that

χ δ ( P n □ P k ) ≥ ω ( ( P n □ P k ) δ ) ≥ ( n − 2 ) ( k − 2 ) 2 .

We note that

( n − 2 ) ( k − 2 ) 2 = ( n − 2 ) ( k − 2 ) 2 if n or k is even , k − 2 2 ( n − 2 ) + n − 2 2 if n and k are odd .

We color V 4 by a coloring c 0 defined by

c 0 ( i , j ) = ( i − 2 ) k − 2 2 + j − 2 2 if i = 2 , … , n − 1 and j = 2 , … , 2 k − 2 2 + 1 , ( n − 2 ) k − 2 2 + i − 2 2 if k is odd and j = k − 1 , i = 2 , … , n − 1 ,

as shown in Figure 5.

The coloring c 0 uses ( n − 2 ) ( k − 2 ) 2 colors. Since the coloring in V 4 has at most two vertices with the same color and they are adjacent in P n □ P k , which are not adjacent in ( P n □ P k ) δ , the coloring c 0 on ( P n □ P k ) δ [ V 4 ] is proper. The case 6 ≤ k ≤ 7 can be verified. Now, we suppose that k ≥ 8 . We color V 3 using 2 n − 2 2 + 2 k − 2 2 colors. We color the vertices V 3 in pair of consecutive vertices (except possibly the last one in a block) clockwise starting from location ( 1 , 2 ) to ( 2 , 1 ) . Each color in V 3 needs to avoid at most 2 n − 2 2 + 2 k − 2 2 − 1 colors of the other vertices in V 3 and two neighbors per each color in V 4 , i.e., we have to avoid 2 n − 2 2 + 2 k − 2 2 + 1 colors. By Lemma 19, there is a remaining color in V 4 that is available to assign to the considered vertex. Since each vertex in V 2 has degree 5 in ( P n □ P k ) δ and ( n − 2 ) ( k − 2 ) 2 > 5 , we can color V 2 . This completes the proof.

$Figure 5 A proper 10-coloring of ( P 6 □ P 7 ) δ {\left({P}_{6}\square {P}_{7})}_{\delta } . The vertices in V 2 {V}_{2} , V 3 {V}_{3} , and V 4 {V}_{4} are shown in red, blue, and black, respectively. The vertices in each V i {V}_{i} for i = 2 , 3 , 4 i=2,3,4 are pairwise adjacent in ( P 6 □ P 7 ) δ {\left({P}_{6}\square {P}_{7})}_{\delta } except for the pairs with a dashed line.$

Figure 5

A proper 10-coloring of ( P 6 □ P 7 ) δ . The vertices in V 2 , V 3 , and V 4 are shown in red, blue, and black, respectively. The vertices in each V i for i = 2 , 3 , 4 are pairwise adjacent in ( P 6 □ P 7 ) δ except for the pairs with a dashed line.

6 Conclusion

In this work, we give a structure of ( G 1 □ ⋯ □ G k ) δ associated with ( G 1 ) δ □ ⋯ □ ( G k ) δ and establish the necessary and sufficient conditions that both graphs are equal. Additionally, we present some lower bounds and upper bounds on the δ -chromatic number for the Cartesian product of graphs and identify some classes of graphs that achieve some of these bounds. We also provide the δ -chromatic number of the Cartesian product of several classes of well-known graphs.

Acknowledgement

The authors are grateful to Rasimate Maungchang and the reviewers for their valuable comments that improved the manuscript.

Funding information: This research project was financially supported by Mahasarakham University.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. WT and PV initiated the project. WT, WP, and PV designed, investigated and proved the results. WT, WP, and PV prepared, reviewed, and edited the manuscript. WT and PV administered the project. All authors have read and agreed to the published version of the manuscript.
Conflict of interest: The authors state no conflict of interest.

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Received: 2024-01-11

Revised: 2024-09-13

Accepted: 2024-10-09

Published Online: 2024-11-18

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

Artikel in diesem Heft

https://doi.org/10.1515/math-2024-0087

Schlagwörter für diesen Artikel

δ-complement graph; chromatic number; Cartesian product; coloring

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