Fast algorithms for Vizing's theorem on bounded degree graphs

Abstract

Vizing's theorem states that every graph G of maximum degree Δ 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 $O\left(m\sqrt{n}\right)$, where $n≔\left|V\right(G\left)\right|$ and $m≔\left|E\right(G\left)\right|$. We investigate the case when Δ is constant, i.e., $\Delta =O\left(1\right)$. In this regime, the runtime of Sinnamon's algorithm is $O\left(n^{3/2}\right)$, which can be improved to $O(n\log n)$, as shown by Gabow, Nishizeki, Kariv, Leven, and Terada. Here we give an algorithm whose running time is only $O\left(n\right)$, which is obviously best possible. Prior to this work, no linear-time $(\Delta +1)$-edge-coloring algorithm was known for any $\Delta ⩾4$. Using some of the same ideas, we also develop new algorithms for $(\Delta +1)$-edge-coloring in the $\mathrm{LOCAL}$ model of distributed computation. Namely, when Δ is constant, we design a deterministic $\mathrm{LOCAL}$ algorithm with running time $\tilde{O}({\log }^{5}n)$ and a randomized $\mathrm{LOCAL}$ algorithm with running time $O({\log }^{2}n)$. Although our focus is on the constant Δ regime, our results remain interesting for Δ up to ${\log }^{o\left(1\right)}n$, since the dependence of their running time on Δ is polynomial. The key new ingredient in our algorithms is a novel application of the entropy compression method.