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 Θ2p-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 Θ2p-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.