Upper bounds for the achromatic and coloring numbers of a graph

Dvořák et al. introduced a variant of the Randić index of a graph G, denoted by R′(G), where R′(G)=∑uv∈E(G)1max{d(u),d(v)}, and 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 χ(G)≤col(G) for any graph G, where χ(G) denotes the chromatic number of G. In this note, we show that for any graph G without isolated vertices, col(G)≤2R′(G), with equality if and only if G is obtained from identifying the center of a star with a vertex of a complete graph. This extends some known results. In addition, we present some new spectral bounds for the coloring and achromatic numbers of a graph.