On Star Coloring of Mycielski's Graphs for Special 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 path of 4 order is bicolored. The minimum number of colors required for a star coloring of G is denoted by χ s (G). Let G is a simple graph with vertex set V (G) = V and edge set E (G) = E, if the vertex set is V ∪ V' ∪ {w} where V′ = {X′:x ∈ V} and the edge set is $E\bigcup {\left\{ {x{y^\prime }:xy \in E} \right\}} \cup \left\{ {{y^\prime }w:{y^\prime } \in {V^\prime }} \right\}$, then G is a Mycielski’s graph. This paper mainly studies star coloring of Mycielski’s graph for some special graphs, the specific results are as follows: if n = 2,3 , then χ s (M (P n )) = 4, otherwise, χ s (M (Pn)) = 6; if n = 5, then χ s (M(C n )) = 8, otherwise, χ s (Μ (C n )) = 6; χ s (Μ (S n )) = 4; χ s (M (K n )) = n + 2.