The paper studies edge-coloring of signed multigraphs and extends classical Theorems of Shannon and König to signed multigraphs. We prove that the chromatic index of a signed multigraph (G,σG) is at most ⌊32Δ(G)⌋. Furthermore, the chromatic index of a balanced signed multigraph (H,σH) is at most Δ(H)+1 and the balanced signed multigraphs with chromatic index Δ(H) are characterized.

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≡2 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.

For 0≤t≤r let m(t,r) be the maximum number s such that every t-edge-connected r-graph has s pairwise disjoint perfect matchings. There are only a few values of m(t,r) known, for instance m(3,3)=m(4,r)=1, and m(t,r)≤r−2 for all t≠5, and m(t,r)≤r−3 if r is even. We prove that m(2l,r)≤3l−6 for every l≥3 and r≥2l.