The complexity of chromatic strength and chromatic edge strength

Abstract

The sum of a coloring is the sum of the colors assigned to the vertices (assuming that the colors are positive integers). The sum ? (G) of graph G is the smallest sum that can be achieved by a proper vertex coloring of G. The chromatic strength s(G) of G is the minimum number of colors that is required by a coloring with sum ? (G). For every k, we determine the complexity of the question Is s(G) ? k?: it is coNP-complete for k = 2 and ?2p-complete for every fixed k ? 3. We also study the complexity of the edge coloring version of the problem, with analogous definitions for the edge sum ??(G) and the chromatic edge strength s?(G). We show that for every k ? 3, it is ?2p-complete to decide whether s?(G) ? k. As a first step of the proof, we present graphs for every r ? 3 with chromatic index r and edge strength r + 1. For some values of r, such graphs have not been known before.