An Improved Sufficient Condition for Reconfiguration of List Edge-Colorings in a Tree

Abstract

SUMMARYWe study the problem of reconfiguring one list edgecoloring of a graph into another list edge-coloring by changing only one edge color assignment at a time, while at all times maintaining a list edgecoloring, given a list of allowed colors for each edge. Ito, Kamiński and Demaine gave a sufficient condition so that any list edge-coloring of a tree can be transformed into any other. In this paper, we give a new sufficient condition which improves the known one. Our sufficient condition is best possible in some sense. The proof is constructive, and yields a polynomial-time algorithm that finds a transformation between two given list edge-colorings of a tree with n vertices via O(n 2 ) recoloring steps. We remark that the upper bound O(n 2 ) on the number of recoloring steps is tight, because there is an infinite family of instances on paths that satisfy our sufficient condition and whose reconfiguration requires Ω(n 2 ) recoloring steps.

The Complexity of (List) Edge-Coloring Reconfiguration Problem

“…Despite recent intensive studies on reconfiguration problems (in particular, for graph colorings [1,2,3,4,5,7,11,12,14,16]), as far as we know, only one complexity result is known for list edge-coloring reconfiguration. Ito et al [11] proved that list edge-coloring reconfiguration is PSPACE-complete even when restricted to k = 6 and planar graphs of maximum degree three.…”

“…(Since the list of each edge is given as an input, this result implies that the problem is PSPACE-complete for every integer k ≥ 6.) They also gave a sufficient condition for which any two list edge-colorings of a tree can be transformed into each other, which was improved by [12]; but these sufficient conditions do not clarify the complexity status for trees, and indeed it remains open.…”

“…Clearly, an (ordinary) edge-coloring of G is a list edge-coloring of G for which L(e) = C holds for every edge e of G, and hence list edge-coloring is a generalization of edge-coloring. Ito et al [12] introduced an adjacency relation defined on list edge-colorings of a graph, and define the list edge-coloring reconfiguration problem, as follows: Suppose that we are given two list edge-colorings of a graph G (e.g., the leftmost and rightmost ones in Fig. 1), and we are asked whether we can transform one into the other via list edge-colorings of G such that each differs from the previous one in only one edge color assignment; such a sequence of list edge-colorings is called a reconfiguration sequence.…”

“…In this paper, we study the reconfiguration problem for (list) edge-colorings of a graph [11,12]. Let C = {1, 2, .…”

Introduction to Reconfiguration

“…Fixed-parameter algorithms have been found when parameterized by k + (where is the length of the reconfiguration sequence) [79,87], parameterized by k and modular-width of the input graph (and hence for cographs when parameterized by k) [75], and for shortest transformation, parameterized by k and the size of the minimum vertex cover (and hence for split graphs parameterized by k) [75]. Other variants for which reconfiguration has been studied include LIST EDGE-COLORING [36,40,100], LIST(2,1)-LABELING [37], CIRCULAR COLORING [101,102], ACYCLIC COLORING [103], and EQUITABLE COLORING [103]. The problem of k-COLORING RECONFIGURATION can also be seen as a special case of HOMOMORPHISM RECONFIGURATION [101,104] and CONSTRAINT SATISFACTION RECONFIGURATION (Section 8).…”