Vizing's theorem for signed multigraphs

Source abstract

We prove that every finite loopless signed multigraph $Σ=(G,σ)$ satisfies $χ'(Σ)\leqΔ(G)+μ(G)$, where $χ'(Σ)$ is its chromatic index, and $Δ(G)$ and $μ(G)$ are the maximum degree and maximum multiplicity of $G$, respectively. This bound is sharp, even when both positive and negative edges are present, and generalizes both Vizing's theorem for ordinary multigraphs and Behr's theorem for signed simple graphs.