On the Independent Coloring of Graphs with Applications to the Independence Number of Cartesian Product Graphs
Nopparat Pleanmani, Sayan Panma, Nuttawoot Nupo · Journal of Mathematics · 2023
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