Improved upper bound for the degenerate and star chromatic numbers of graphs

Let be a graphLet G = G(V, E) be a graph. A proper coloring of G is a function f : V -> N such that f (x) not equal f (y) for every edge xy is an element of E. A proper coloring of a graph G such that for every k >= 1, the union of any k color classes induces a (k - 1)-degenerate subgraph is called a degenerate coloring; a proper coloring of a graph with no two-colored P-4 is called a star coloring. If a coloring is both degenerate and star, then we call it a degenerate star coloring of graph. The corresponding chromatic number is denoted as chi(sd) (G). In this paper, we employ entropy compression method to obtain a new upper bound chi(sd) (G) = <= inverted right perpendicular19/6 Delta(3/2) + 5 Delta inverted left perpendicular for general graph G.. A proper coloring of G is a function such that for every edge . A proper coloring of a graph G such that for every , the union of any k color classes induces a -degenerate subgraph is called a degenerate coloring; a proper coloring of a graph with no two-colored is called a star coloring. If a coloring is both degenerate and star, then we call it a degenerate star coloring of graph. The corresponding chromatic number is denoted as . In this paper, we employ entropy compression method to obtain a new upper bound for general graph G.