Randic index and coloring number of a graph

The Randic index of a graph G, denoted by R(G), is defined as the sum of 1/root d(u)d(v) for all edges uv of G, where d(u) denotes the degree of a vertex u in G. The coloring number col(G) of a graph G is the smallest number k for which there exists a linear ordering of the vertices of G such that each vertex is preceded by fewer than k of its neighbors. It is well-known that chi (G) <= col(G) for any graph G, where chi (G) denotes the chromatic number of G. In this note, we show that for any graph G with at least one edge, col(G) <= 2R(G), with equality if and only if G is a complete graph, possibly with some additional isolated vertices. This extends a theorem of Hansen and Vukicevic. (C) 2014 Elsevier B.V. All rights reserved.