Abstract
Let be a graph with . A nonempty subset of is called an independent set of if no two distinct vertices in are adjacent. The union of a class { is an independent set of } and is denoted by . For a graph , a function is called an independent coloring of (or simply called an coloring) if for any adjacent vertices and is a class of disjoint sets. Let denote the maximum cardinality of the set{ is an coloring of }. In this paper, we obtain basic properties of an coloring of and find of some families of graphs and . Furthermore, we apply them to determine the independence number of the Cartesian product of a complete graph and a graph and prove that .
1. Introduction
In graph theory, the study of graph invariants is a worldwide research related to a property of graphs that depends on the abstract structure. This invariant property can be formalized as a function from graphs to a class of values such that any two isomorphic graphs have the same value. Two well-known graph invariants or parameters are the independence number and the chromatic number of graphs which indicate the maximum size and the minimum size of independent sets and color sets of graphs, respectively. Their applications can be found in several fields such as computer science, engineering, and optimization problems. Many researchers study those parameters in various ways and sometime combine their concepts together to model certain new structures. For example, in 2011, Arumugam et al. [1] investigated maximal independent sets in minimum colorings. In 2012, Wu and Hao [2] presented an effective approach to coloring large graphs by using a preprocessing method to extract large independent sets from the graphs. In 2016, Samanta et al. [3] introduced a new concept of coloring of fuzzy graphs to color world political map mentioning the strength of relationship among the countries. Later in 2020, Mahapatra et al. [4] proposed the edge coloring of fuzzy graphs to solve job oriented web sites and traffic light problems. Recently, in 2022, Brešar and Štesl [5] presented the independence coloring game of graphs and proved that the independence game chromatic number of a tree can be arbitrarily large. In this paper, we introduce a new parameter of graphs, called the independent coloring, which is motivated from the combination of the independence and coloring concepts. Moreover, prominent properties of the independent coloring are provided. Furthermore, we apply the result for studying the independence number of Cartesian product graphs. More information about other operations and product of graphs can be found in [6–9].
2. Preliminaries and Notations
Throughout this paper, all graphs are considered to be finite and simple. Let be a graph of order . Two vertices and are said to be adjacent, if . For a nonempty subset of , the induced subgraph of induced by , denoted by , is a subgraph of provided that if and , then , as well. A subgraph of is said to be spanning if . We remark that a graph without edges is called an empty graph. For a non-negative integer , we denote , , , , and to be a complete graph of order , a cycle of length , a path of length , a wheel of order , and a star of order , respectively. For other graph terminologies and notations, we refer the reader to [10].
A set of vertices of is said to be an independent set of if no two distinct vertices in are adjacent. The maximum cardinality of an independent set of is called the independence number of and is denoted by . If is an independent set such that , we say that is an set of . For a positive integer , an coloring of a graph means a surjection from to the set with for every adjacent vertices in . The chromatic number of is defined to be the minimum of over all colorings of , and we denote a coloring of by coloring of .
We now introduce a graph parameter. For a graph , we denote by the class { which is an independent set of } . Let and be graphs with . A function is called an independent coloring of (or simply called an coloring) if for any adjacent vertices and is a class of disjoint sets. An coloring of is said to be trivial when for any vertex . In addition, we denote by the maximum cardinality of the set { is an coloring of }. If is an coloring of such that , we say that is an coloring of . It is not hard to see that . For a trivial case , we note that is actually the independence number of .
3. Independent Coloring of Graphs
We start this section by presenting the following two useful lemmas which are referred in the proofs of other results.
Lemma 1. Let and be graphs and let be an coloring of with , where . Then, the following statements hold:(1) is an independent set of for each .(2) and are disjoint for .(3).
Proof. Let . If there are adjacent vertices , then , a contradiction. Hence, (1) holds. Next, we show that (2) holds. It is clear for . Then, we may assume that . Let with . If there is a vertex , then . Therefore, and this implies , a contradiction. Hence, (2) holds. And this leads toand hence (3) holds.
Lemma 2. Let and be graphs and let be an -coloring of . If is a function defined byfor all , then the following statements hold:(1) is a coloring of .(2).
Proof. It is clear, by Lemma 1, that (1) holds. Let where . Again by Lemma 1, we obtain thatas required.
We note that graphs and in can be switched as in the following result.
Proposition 3. Let and be graphs. Then,
Proof. Let be an coloring of and be a coloring of defined in Lemma 2. Then, by Lemma 2, . Similarly, . Hence, the equality follows.
Proposition 4. Let , , and be graphs. Then, if is a spanning subgraph of .
Proof. Let be an coloring of . Define by for all . Clearly, is an coloring of . Hence, .
We now give some basic properties of an coloring of which are useful for describing the lower bound and the upper bound of .
Lemma 5. Let and be graphs and be an coloring of .(1)If , then is a partition of .(2)If and , then .
Proof. Assume that . Let with . We show that . Suppose, to the contrary, that there is a vertex . Define byfor all . Obviously, is an coloring of . However,which is impossible. Therefore, . Hence, is a partition of .
Next, suppose that and . Let be an set of and with . If , then define byfor all . It is easy to see that is an coloring of . However,which is impossible. Hence, .
We obtain the lower and upper bounds for in terms of independence numbers and order of graphs.
Theorem 6. Let and be graphs. Then,
Proof. Let be an set of and be an set of . Without loss of generality, we can assume that . We consider the following two cases.
Case 1. .
Define byfor all . Clearly, is an coloring of . Thus,
Case 2. .
Let . Define byfor all . Clearly, is an coloring of . Thus,Next, let be an coloring of and be an coloring of with and , where and . By Lemma 1, it follows thatand similarly,Therefore, . Consequently, the result follows.
To establish the sharpness of the lower bound stated in Theorem 6, consider an empty graph . Obviously, . Next, we investigate the sharpness of the upper bound also stated in Theorem 6. Let and , where with . We see that .
Lemma 9 (see [10]). Let be any graph. Then, .
Proof. Let be a coloring of . Then, .
Corollary 10. Let and be graphs. Then,
Proof. By Theorem 6 and Lemma 9, we can conclude thatThe Nordhaus–Gaddum bound is a sharp lower or upper bound on the sum or product of a parameter of a graph and its complement. By deriving of Corollary 10, we obtain sharp bounds for in terms of , , and . Moreover, the graph families attaining the bounds in Proposition 3 attain these bounds also.
Corollary 11. For every graph ,
Now, we focus on a coloring of . Before that, we need the following lemma.
Lemma 12. For a positive integer , let be real numbers. Then,
Proof. For , we obviously have . Consequently,Therefore, .
The following theorem shows that is a sum of squares of positive integers.
Theorem 13. For any graph , we have
Proof. Let . Furthermore, let be a class of disjoint independent sets of such that , where is a positive integer. We first show that . Define byfor all . Clearly, is a coloring of . And thus, by Lemma 1,Now, we show that . Let be an coloring of . Furthermore, let be such that , where is a positive integer. By Lemmas 1 and 12, we obtain that Case 1. . Clearly, . Since both of and are classes of disjoint independent sets of , we obtain that Case 2. . Since both of and are classes of disjoint independent sets of , we get that Hence, the equality holds. Next, we consider the parameter in case one of two graphs is complete.
Theorem 14. For any graph and a positive integer , let { be a class of disjoint independent sets of with }. Then,
Proof. Let . Case 1. . Since , we have . By Lemma 9, we get . Therefore, by Theorem 6, . Next, let be a coloring of and , where . Define by for all . It is clear that is a coloring of . This implies, by Lemma 1, that Hence, , as required. Case 2. . Let be a coloring of . Claim . Suppose, to the contrary, that . By Lemma 5, is a partition of . Thus,a contradiction. This proves our claim. Since every independent set of is singleton and by Lemma 1, we get which is a class of disjoint independent sets of . Let . It follows, by Lemma 1, thatNext, we show that . Let be a class of disjoint independent sets of with such that . Define byfor all . Clearly, is a coloring of . This implies, by Lemma 1, that . Hence, , as required.
The following corollaries present the prominent results of in which and are elements of some basic families of graphs.
Corollary 15. Let be a graph and be a positive integer. If is bipartite, then
Corollary 16. Let be a graph and be a positive integer greater than 1. Then, the following statements hold:(1) for all non-negative integers .(2) for all non-negative integers .(3) for all positive integers .(4) for all positive integers .
Corollary 17. For positive integers , we have
Proof. Without loss of generality, let . For an injection , we see that is a coloring of , since every independent set of is singleton. This implies thatHence, the result follows by Theorem 6.
Corollary 18. For positive integers with , we have
Proof. Let be a coloring of such that and . Case 1. . Let . Then, we define a function by for all . It is precise that is a coloring of and so Therefore, by Theorem 6, the result follows. Case 2. . Let . Define a function by for all . It is easy to show that is a coloring of , since every independent set of is singleton. This implies thatHence, the result follows by Theorem 6.
Corollary 19. For positive integers with , we have
Proof. Let be a coloring of such that and . Case 1. . Let . Define a function by for all . Clearly, is a coloring of . Then, By Theorem 6, we obtain that . Case 2. . Let . Define a function by for all . Clearly, is a coloring of . Then, By Theorem 6, we get that . Case 3. . Let . Define a function by for all . Clearly, is a coloring of . Then, By Theorem 6, we have . This completes the proof.
Corollary 20. For positive integers with , we have
Proof. By applying the proof of Corollary 19, the result follows.
Actually, we can determine , where and belong to some basic families of graphs by applying the proofs of Corollaries 15, 16, and 19. As shown in Table 1, we proved only the value in the first column. However, we determine the rest without proofs and we leave the rest to the reader as an exercise.
4. Applications on Cartesian Product Graphs
In this section, we apply the coloring by independent sets to the Cartesian product of a complete graph and a graph. Firstly, we provide some preparations for background.
Given two graphs and , many definitions exist that are known as the product of and . For a detailed treatment of graph products, we refer the reader to [11, 12]. The Cartesian product of and , denoted by , is the graph with vertex set , where two vertices and are adjacent whenever and , or and . There are several types of graphs defined by the Cartesian product of graphs. In particular, we focus the following types of those graphs. For a positive integer , an ladder graph is defined to be the Cartesian product graph . An book graph means the Cartesian product graph . An dimensional hypercube is recursively defined to be the Cartesian product graph , where and . And we note that and which can be found in [13].
In order to properly study graph products, we need some definitions that consider the set product of sets and . In particular, for , we denote by { where }. Moreover, for , we denote by .
Determining the independence number and its variants of a graph product in terms of its factors is well studied in graph theory. For papers concerning the graph products, we refer the reader, for example, to [14–18]. In this section, we continue the study of the graph product independence by considering the independence of the Cartesian product graphs. Namely, this section provides results regarding the independence number and gives certain valuable corollaries to the results.
Now, we characterize the independent sets of the Cartesian product of a complete graph and a graph.
Theorem 21. Let be a graph. For a positive integer , let be a nonempty subset of . Then, is an independent set of if and only if the following conditions hold:(1) is an independent set of for every .(2) for any adjacent vertices .
Proof. Let be an independent set of .
Furthermore, let and . If , then . So, we assume the rest that . Since , we have . Thus, .
Let be two adjacent vertices in . Furthermore, let and . Since , we have . Thus, since . Hence, . For the converse, we assume that (1) and (2) hold. Let . If , then clearly . We obtain that because . Thus, . If , then we distinguish the following two cases. Case 1. . Clearly, . Thus, , since is an independent set of . Therefore, . Case 2. . It is easy to see that , since neither nor .
Theorem 22. Let be a graph. For a positive integer , we have
Proof. We first show that . Let be an set of . Define a function byfor all . By Theorem 21, we obtain that is an coloring of . Consequently,Next, we show that . Let be an coloring of . Furthermore, let and . We see that is independent in for every and for every adjacent vertices in . It follows from Theorem 21 that is an independent set of for every and so is . Consequently,Hence, the result follows.
Corollary 23. For a positive integer , we have
Proof. Since and by Theorem 22, we get that . By Corollary 15, we have because .
Corollary 24. For a positive integer , we have
Proof. Since and by Theorem 22, we obtain that . Therefore, by Corollary 15, we conclude that , since .
Corollary 25 (see [13]). For a positive integer , we have
Proof. It is clear for . So, we assume that . Since and by Theorem 22, we get . Since and by Corollary 15, we have .
5. Conclusion
The concept of the independent coloring of graphs has been introduced in this paper. Prominent properties and results of the independent coloring have been proposed. Especially, the values of , where and are fundamental graphs, have been collected and presented in the table. Finally, the independent coloring has been applied to determine the independence number of Cartesian product graphs. Actually, the independent coloring of graphs can be considered as a generalized concept of the independence number of graphs, as well.
Data Availability
No data were used in this research.
Conflicts of Interest
The authors declare that they have no conflict of interest.
Acknowledgments
This research was supported by the Department of Mathematics, Faculty of Science, Khon Kaen University, Fiscal Year 2022.