The Complexity of (List) Edge-Coloring Reconfiguration Problem

摘要

Let $G$ be a graph such that each edge has its list of available colors, andassume 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 $G$, andasked 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 theprevious one by changing a color assignment of exactly one edge. This problemis known to be PSPACE-complete for every integer $k \ge 6$ and planar graphs ofmaximum degree three, but any complexity hardness was unknown for the non-listvariant. In this paper, we first improve the known result by proving that, forevery integer $k \ge 4$, the problem remains PSPACE-complete even if an inputgraph is planar, bounded bandwidth, and of maximum degree three. We then givethe first complexity hardness result for the non-list variant: for everyinteger $k \ge 5$, we prove that the non-list variant is PSPACE-complete evenif an input graph is planar, of bandwidth linear in $k$, and of maximum degree$k$.