The Complexity of Counting Edge Colorings for Simple Graphs

Jin-Yi Cai; Artem Govorov

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arXiv · 2010.04910

The Complexity of Counting Edge Colorings for Simple Graphs

Jin-Yi Cai·Artem Govorov

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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 $(\kappa, r)$.

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Jin-Yi Cai, Artem Govorov. 2020-10-10. The Complexity of Counting Edge Colorings for Simple Graphs. https://arxiv.org/abs/2010.04910

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