On Star Coloring of Several Corona Graphs

Abstract

Let G be a simple graph with vertex set V(G) and edge set E(G). A vertex coloring of G is called a star coloring of G if any of the paths of 4 order are bicolored. The minimum number of colors required for a star coloring of G is denoted by χs (G). The corona product of simple graphs G of order m and H of order n is graph G ∘ H with vertex set V(G ∘ H) = {vi |i = 1,2,⋯m}∪{vij |i = 1,2,⋯m, j = 1,2,⋯n}, in which vi is adjacent to every vertex of Hi if and only if, vi ∈ V(G), vij ∈ V(Hi ). According to the existing graph dyeing literature, it has become a very important technical means to study the graph dyeing problem by using the graph structure operation. Therefore, it is of great significance to study the star coloring of graphs for studying the acyclic coloring and distance coloring of graphs, the study has strong application background and great theoretical value for computing graphs. In this paper, we find the upper bound of χs (G ∘ H) and the exact values of χs (G ∘ H) of the corona product G ∘ H of two graphs G and H as: χs (G ∘ H) ≤ χs (G) + χs (H); χs (Pm ∘ H) = χs (H) + 2; χs (K 1,m ∘ H) = χs (H) + 2; χs (Cn ∘ H) = χs (H) + 2, where n ≠ 5.