The Complexity Of Counting Edge Colorings For Simple Graphs

We prove #P-completeness results for counting edge colorings on simplegraphs. These strengthen the corresponding results on multigraphs from[4]. We prove that for any kappa >= r >= 3 counting kappa-edge colorings on r-regular simple graphs is #P-complete. Furthermore, we show that for planar r-regular simple graphs, where r is an element of {3, 4, 5}, counting edge colorings with.colors for any kappa >= r is also #P-complete. As there are no planar r-regular simple graphs for any r> 5, these statements cover all interesting cases in terms of the parameters (kappa, r). (C) 2021 Elsevier B.V. All rights reserved.