The Complexity of Counting Edge Colorings for Simple Graphs

Abstract

We prove #P-completeness results for counting edge colorings on simple graphs. These strengthen the corresponding results on multigraphs from [4]. We prove that for any $\kappa \ge r\ge 3$ counting $\kappa$-edge colorings on $r$-regular simple graphs is #P-complete. Furthermore, we show that for planar $r$-regular simple graphs where $r\in \{3,4,5\}$ counting edge colorings with \k{appa} colors for any $\kappa \ge r$ 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 $(\kappa ,r)$.