The complexity of chromatic strength and chromatic edge strength

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 p 2-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 p 2-complete to decide whether s (G) k holds. As a first step of the proof, we present

• Type: Journal
• Year: 2006
• Where: CC
• Authors: Dániel Marx
• Keywords: CC 2006, Positive Integers, Proper Vertex, Smallest Sum, System Software