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

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