The Complexity of (List) Edge-Coloring Reconfiguration Problem

Abstract

Let G be a graph such that each edge has its list of available colors, and assume that each list is a subset of the common set consisting of k colors. Suppose that we are given two list edge-colorings \(f_0\) and \(f_r\) of G, and asked whether there exists a sequence of list edge-colorings of G between \(f_0\) and \(f_r\) such that each list edge-coloring can be obtained from the previous one by changing a color assignment of exactly one edge. This problem is known to be PSPACE-complete for every integer \(k \ge 6\) and planar graphs of maximum degree three, but any computational hardness was unknown for the non-list variant in which every edge has the same list of k colors. In this paper, we first improve the known result by proving that, for every integer \(k \ge 4\), the problem remains PSPACE-complete even for planar graphs of maximum degree three and bounded bandwidth. Since the problem is known to be solvable in polynomial time if \(k \le 3\), our result gives a sharp analysis of the complexity status with respect to the number k of colors. We then give the first computational hardness result for the non-list variant: for every integer \(k \ge 5\), the non-list variant is PSPACE-complete even for planar graphs of maximum degree k and bandwidth linear in k.