Pairwise disjoint perfect matchings in $r$-edge-connected $r$-regular graphs

TL;DR

This paper investigates the existence of pairwise disjoint perfect matchings in highly edge-connected regular graphs, resolving Thomassen's problem for even degrees and linking it to major conjectures in cubic graph theory.

Contribution

It proves Thomassen's conjecture for all even degrees except those congruent to 2 mod 4, and establishes equivalences relating perfect matchings to well-known conjectures in cubic graphs.

Findings

Thomassen's conjecture is false for r ≡ 2 mod 4.

Resolved Thomassen's problem for all even r except those ≡ 2 mod 4.

Connected perfect matchings in 5-regular graphs to major cubic graph conjectures.

Abstract

Thomassen [Problem 1 in Factorizing regular graphs, J. Combin. Theory Ser. B, 141 (2020), 343-351] asked whether every $r$-edge-connected $r$-regular graph of even order has $r-2$ pairwise disjoint perfect matchings. We show that this is not the case if $r \equiv 2 \text{ mod } 4$. Together with a recent result of Mattiolo and Steffen [Highly edge-connected regular graphs without large factorizable subgraphs, J. Graph Theory, 99 (2022), 107-116] this solves Thomassen's problem for all even $r$. It turns out that our methods are limited to the even case of Thomassen's problem. We then prove some equivalences of statements on pairwise disjoint perfect matchings in highly edge-connected regular graphs, where the perfect matchings contain or avoid fixed sets of edges. Based on these results we relate statements on pairwise disjoint perfect matchings of 5-edge-connected 5-regular graphs to well-known conjectures for cubic graphs, such as the Fan-Raspaud Conjecture, the Berge-Fulkerson Conjecture and the $5$-Cycle Double Cover Conjecture.