Fast algorithms for Vizing's theorem on bounded degree graphs

Vizing's theorem states that every graph $G$ of maximum degree $\Delta$ can be properly edge-colored using $\Delta + 1$ colors. The fastest currently known $(\Delta+1)$-edge-coloring algorithm for general graphs is due to Sinnamon and runs in time $O(m\sqrt{n})$, where $n = |V(G)|$ and $m =|E(G)|$. Using the bound $m \leq \Delta n/2$, the running time of Sinnamon's algorithm can be expressed as $O(\Delta n^{3/2})$. In the regime when $\Delta$ is considerably smaller than $n$ (for instance, when $\Delta$ is a constant), this can be improved, as Gabow, Nishizeki, Kariv, Leven, and Terada designed an algorithm with running time $O(\Delta m \log n) = O(\Delta^2 n \log n)$. Here we give an algorithm whose running time is only linear in $n$ (which is obviously best possible) and polynomial in $\Delta$. We also develop new algorithms for $(\Delta+1)$-edge-coloring in the $\mathsf{LOCAL}$ model of distributed computation. Namely, we design a deterministic $\mathsf{LOCAL}$ algorithm with running time $\mathsf{poly}(\Delta, \log\log n) \log^5 n$ and a randomized $\mathsf{LOCAL}$ algorithm with running time $\mathsf{poly}(\Delta) \log^2 n$. The key new ingredient in our algorithms is a novel application of the entropy compression method.