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

Abstract

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. The 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 find 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 .