The Complexity of Counting Edge Colorings for Simple Graphs

TL;DR

This paper establishes that counting edge colorings in simple graphs is computationally #P-complete across various classes, including regular and planar graphs, extending previous multigraph results.

Contribution

It proves #P-completeness for counting edge colorings in simple graphs, including planar and regular cases, strengthening prior multigraph results.

Findings

Counting edge colorings is #P-complete for simple graphs.

The results cover all relevant parameters for regular and planar graphs.

Extends complexity results from multigraphs to 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 $(\kappa, r)$.