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 f₀ and f_r of G, and asked whether there exists a sequence of list edge-colorings of G between f₀ 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 ≥ 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 ≥ 4, the problem remains PSPACE-complete even for planar graphs of bounded bandwidth and maximum degree three. Since the problem is known to be solvable in polynomial time if k ≤ 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 ≥ 5, the non-list variant is PSPACE-complete even for planar graphs of bandwidth quadratic in k and maximum degree k.