Upper bound for DP-chromatic number of a graph

Abstract

DP-coloring as a generalization of list coloring was introduced recently by Dvořák and Postle. The D P -chromatic number of G , denoted by χ D P ( G ) , is the minimum number k such that G is D P − k -colorable. The variation of the Randić index R ′ ( G ) of a graph G is defined as the sum of the weights 1 m a x { d ( u ) , d ( v ) } of all edges u v of G , where d ( u ) is the degree of the vertex u in G . In this paper, we show that for any graph G of order n , χ D P ( G ) ≤ 2 R ′ ( G ) , and this bound is sharp for all n and 2 ≤ χ D P ( G ) ≤ n .