Finding Reconfigurations between List Edge-Colorings of a Graph

Reconfiguration problems arise when we wish to find a step-by-step transformation between two feasible solutions of a problem such that all intermediate results are also feasible. Recently, Ito et al. [5] proposed a framework of reconfiguration problems, and gave complexity and approximability results for reconfiguration problems derived from several well-known problems, such as independent set, clique, matching, etc. In this paper, we study a reconfiguration problem for list edge-colorings of a graph. An (ordinary) edge-coloring of a graph G is an assignment of colors from a color set C to each edge of G so that every two adjacent edges receive different colors. In list edge-coloring, each edge e of G has a set L(e) of colors, called the list of e. Then, an edge-coloring f of G is called an L-edge-coloring of G if f(e) ∈ L(e) for each edge e, where f(e) denotes the color assigned to e by f . Fig. 1 illustrates three L-edge-colorings of the same graph with the same list L; the color assigned to each edge is surrounded by a box in the list. Clearly, an edge-coloring is an L-edge-coloring for which L(e) is the same color set C for every edge e of G, and hence list edge-coloring is a generalization of edge-coloring. Suppose now that we are given two L-edge-colorings of a graph G (e.g., the ones in Fig. 1(a) and (c)), and we are asked whether we can transform one into the other via L-edge-colorings of G such that each differs from the previous one in only one edge color assignment. We call this problem the list edge-coloring reconfiguration problem. For the particular instance of Fig. 1, the answer is “yes,” as illustrated in Fig. 1, where the edge whose color assignment was changed from the previous one is de-