On Star Coloring of Several Corona Graphs

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 χ s ( 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 ) = { v i | i = 1,2,⋯ m }∪{ v ij | i = 1,2,⋯ m , j = 1,2,⋯ n }, in which v i is adjacent to every vertex of H i if and only if, v i ∈ V ( G ), v ij ∈ V ( H i ). 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 χ s ( G ∘ H ) and the exact values of χ s ( G ∘ H ) of the corona product G ∘ H of two graphs G and H as: χ s ( G ∘ H ) ≤ χ s ( G ) + χ s ( H ); χ s ( P m ∘ H ) = χ s ( H ) + 2; χ s ( K 1, m ∘ H ) = χ s ( H ) + 2; χ s ( C n ∘ H ) = χ s ( H ) + 2, where n ≠ 5.