On the Independent Coloring of Graphs with Applications to the Independence Number of Cartesian Product Graphs

. Let G be a graph with V � V ( G ) . A nonempty subset S of V is called an independent set of G if no two distinct vertices in S are adjacent. Te union of a class { S : S is an independent set of G } and ∅ { } is denoted by I ( G ) . For a graph H , a function f : V ⟶ I ( H ) is called an H − independent coloring of G (or simply called an H − coloring) if f ( x ) ∩ f ( y ) � ∅ for any adjacent vertices x, y ∈ V and f ( V ) is a class of disjoint sets. Let α ( H, G ) denote the maximum cardinality of the set{ 􏽐 x ∈ V | f ( x )| : f is an H − coloring of G }. In this paper, we obtain basic properties of an H − coloring of G and fnd α ( H, G ) of some families of graphs G and H . Furthermore, we apply them to determine the independence number of the Cartesian product of a complete graph K n and a graph G and prove that α ( K n □ G ) � α ( K n , G ) .


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.Tis 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 wellknown 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.Teir applications can be found in several felds 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 efective 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 trafc 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][7][8][9].

Preliminaries and Notations
Troughout this paper, all graphs are considered to be fnite and simple.Let G � (V, E) be a graph of order |V|.Two vertices u and v are said to be adjacent, if uv ∈ E. For a nonempty subset S of V, the induced subgraph of G induced by S, denoted by G[S], is a subgraph of G provided that if u, v ∈ S and uv ∈ E, then uv ∈ E(G[S]), as well.A subgraph H of G is said to be spanning if V(H) � V(G).We remark that a graph without edges is called an empty graph.For a non-negative integer n, we denote K n , C n , P n , W n , and S n to be a complete graph of order n ≥ 1, a cycle of length n ≥ 0, a path of length n ≥ 0, a wheel of order n ≥ 4, and a star of order n ≥ 2, respectively.For other graph terminologies and notations, we refer the reader to [10].
A set S of vertices of G is said to be an independent set of G if no two distinct vertices in S are adjacent.Te maximum cardinality of an independent set of G is called the independence number of G and is denoted by α(G).If S is an independent set such that |S| � α(G), we say that S is an α − set of G.For a positive integer n, an n− coloring of a graph G means a surjection from V(G) to the set 1, 2, . . ., n { } with f(u) ≠ f(v) for every adjacent vertices u, v in G. Te chromatic number χ(G) of G is defned to be the minimum of i over all i− colorings of G, and we denote a χ(G)− coloring of G by χ− coloring of G.
We now introduce a graph parameter.For a graph G, we denote by I(G) the class {S: S which is an independent set of G} ∪ ∅ { }.Let G and H be graphs with

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 G and H be graphs and let f be an
Ten, the following statements hold: holds.And this leads to and hence (3) holds.

□
Lemma 2. Let G and H be graphs and let f be an for all x ∈ V(H), then the following statements hold: as required.
We now give some basic properties of an α − H− coloring of G which are useful for describing the lower bound and the upper bound of α(H, G). □ Lemma 5. Let G and H be graphs and f be an α − H− coloring of G.
for all x ∈ V(G).It is easy to see that g is an H− coloring of G.However, We obtain the lower and upper bounds for α(H, G) in terms of independence numbers and order of graphs.□

Journal of Mathematics
Theorem 6.Let G and H be graphs.Ten, Proof.Let A be an α− set of G and B be an α− set of H. Without loss of generality, we can assume that We consider the following two cases.
Next, let f be an α − H− coloring of G and g be an and similarly, Consequently, the result follows.
To establish the sharpness of the lower bound stated in Teorem 6, consider an empty graph H. Obviously, Lemma 9 (see [10]).Let G be any graph.Ten, Proof.By Teorem 6 and Lemma 9, we can conclude that 4 Journal of Mathematics Te 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 α(G)α(G) in terms of α(G, G), χ(G), and χ(G).Moreover, the graph families attaining the bounds in Proposition 3 attain these bounds also.

□ Corollary 11. For every graph G,
Now, we focus on a G − coloring of G. Before that, we need the following lemma.

Lemma
12. For a positive integer n, let x 1 , x 2 , . . ., x n , y 1 , y 2 , . . ., y n be real numbers.Ten, (20) , where m is a positive integer.By Lemmas 1 and 12, we obtain that Hence, the equality holds.Next, we consider the parameter in case one of two graphs is complete.

□ Theorem 14. For any graph G and a positive integer n, let
Suppose, to the contrary, that ∅∈f( a contradiction.Tis proves our claim.Since every independent set of K n is singleton and by Lemma 1, we get Te following corollaries present the prominent results of α(H, G) in which G and H are elements of some basic families of graphs.

□ Corollary 15. Let G be a graph and n
Corollary 16.Let G be a graph and n be a positive integer greater than 1.Ten, the following statements hold: (1) α(P 2m , K n ) � 2m + 1 for all non-negative integers m.

Corollary 17.
For positive integers m and n, we have Proof.Without loss of generality, let m ≤ n.For an injection Hence, the result follows by Teorem 6.

Journal of Mathematics
Hence, the result follows by Teorem 6.

□
Corollary 19.For positive integers m, n with m, n ≥ 2, we have By Teorem 6, we obtain that By Teorem 6, we get that  1, we proved only the value α(H, G) in the frst column.However, we determine the rest without proofs and we leave the rest to the reader as an exercise.

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 G and H, many defnitions exist that are known as the product of G and H.For a detailed treatment of graph products, we refer the reader to [11,12].Te Cartesian product of G and H, denoted by G□H, is the graph with vertex set V(G) × V(H), where two vertices (v 1 , h 1 ) and Tere are several types of graphs defned by the Cartesian product of graphs.In particular, we focus the following types of those graphs.For a positive integer n, an n− ladder graph L n is defned to be the Cartesian product graph K 2 □P n .An n − book graph B n means the Cartesian product graph K 2 □S n+1 .An n− dimensional hypercube Q n is recursively defned to be the Cartesian product graph K 2 □Q n− 1 , where n ≥ 2 and Q 1 � K 2 .And we note that |V(Q n )| � 2 n and χ(Q n ) � 2 which can be found in [13].
In order to properly study graph products, we need some defnitions that consider the set product of sets A and B. In particular, for S ⊆ A × B, we denote by π 1 (S) � {a: (a, b) ∈ S where b ∈ B}.Moreover, for s ∈ π 1 (S), we denote by π s (S) � b: (s, b) ∈ S { }.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][15][16][17][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 H be a graph.For a positive integer n, let S be a nonempty subset of V(K n □H).Ten, S is an independent set of K n □H if and only if the following conditions hold: (1) π s (S) is an independent set of H for every s ∈ π 1 (S).
Proof.Let S be an independent set of K n □H.Furthermore, let s ∈ π 1 (S) and u s , v s ∈ π s (S).If u s � v s , then u s v s ∉ E(H).So, we assume the rest that u s ≠ v s .Since (s, u s ), (s, v s ) ∈ S, we have (s, u s )(s, v s ) ∉ E(K n □H).Tus, u s v s ∉ E.
Let a, b be two adjacent vertices in π 1 (S).Furthermore, let v a ∈ π a (S) and For the converse, we assume that (1) and ( 2

□
Theorem 22.Let H be a graph.For a positive integer n, we have where G, H belong to some basic families of graphs.

Conclusion
Te 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 α(H, G), where H and G 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.

􏽐 n i�1 y 2 iTheorem 13 .
.Te following theorem shows that α(G, G) is a sum of squares of positive integers.□For any graph G, we haveα(G, G) � max  P∈P |P| 2 : P is a class of disjoint independent sets of G Let M � max  P∈P |P| 2 :  P is a class of disjoint independent sets of G}.Furthermore, let P ′ � D 1 ,  D 2 , . . ., D m } be a class of disjoint independent sets of G such that  P∈P ′ |P| 2 � M,where m is a positive integer.We frst show that α(G, G) ≥ M. Defne g: V(G) ⟶ I(G) by g(x) � P, if x ∈ P for some P ∈ P ′ , for all x ∈ V(G).Clearly, g is a G− coloring of G.And thus, by Lemma 1, For a positive integer n, we haveα Q n  � 2 n− 1 .(55)Proof.It is clear for n � 1.So, we assume that n ≥ 2. Since Q n � K 2 □Q n− 1 and by Teorem 22, we get α(Q n ) � α(K 2 , Q n− 1 ).Since χ(Q n− 1 ) � 2 and by Corollary 15, we have α(Q n ) � |V(Q n− 1 )| � 2 n− 1 .
we investigate the sharpness of the upper bound also stated in Teorem 6.Let G � C 2m and H � C 2n , where m, n∈ N with m, n > 1.We see that α Proof.By applying the proof of Corollary 19, the result follows.Actually, we can determine α(H, G), where G and H belong to some basic families of graphs by applying the proofs of Corollaries 15, 16, and 19.As shown in Table