Fast algorithms for Vizing's theorem on bounded degree graphs - Bernshteyn, Anton; Dhawan, Abhishek (Journal of combinatorial theory. Series B, 2025)
Vizing's theorem states that every graph G of maximum degree Δ can be properly edge-colored using Δ+1 colors. The fastest currently known (Δ+1)-edge-coloring algorithm for general graphs is due to Sinnamon and runs in time O(mn), where n≔|V(G)| and m≔|E(G)|. We investigate the case when Δ is constant, i.e., Δ=O(1). In this regime, the runtime of Sinnamon's algorithm is O(n3/2), which can be improved to O(nlog⁡n), as shown by Gabow, Nishizeki, Kariv, Leven, and Terada. Here we give an algorithm whose running time is only O(n), which is obviously best possible. Prior to this work, no linear-time (Δ+1)-edge-coloring algorithm was known for any Δ⩾4. Using some of the same ideas, we also develop new algorithms for (Δ+1)-edge-coloring in the LOCAL model of distributed computation. Namely, when Δ is constant, we design a deterministic LOCAL algorithm with running time O˜(log5⁡n) and a randomized LOCAL algorithm with running time O(log2⁡n). Although our focus is on the constant Δ regime, our results remain interesting for Δ up to logo(1)⁡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.