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 complexity hardness was unknown for the non-list variant. In this paper, we first improve the known result by proving that, for every integer $k \ge 4$, the problem remains PSPACE-complete even if an input graph is planar, bounded bandwidth, and of maximum degree three. We then give the first complexity hardness result for the non-list variant: for every integer $k \ge 5$, we prove that the non-list variant is PSPACE-complete even if an input graph is planar, of bandwidth linear in $k$, and of maximum degree $k$.