A Fast Deterministic Algorithm for $(\Delta+1)$-edge coloring in CONGEST
TL;DR
The first $poly(\Delta,\log n)-round algorithm for $(\Delta + 1)$-edge coloring in the CONGEST model is presented and the $n$-dependency of its runtime, $\tilde{O}(\log^5 n)$, matches the best published dependency in the LOCAL model.
Abstract
Vizing's theorem states that any graph of maximum degree $\Delta$ can be properly edge-colored with $\Delta + 1$ colors (which is optimal in general). A recent breakthrough result by Bernshteyn showed that such a $(\Delta + 1)$-edge coloring can be found deterministically in $poly(\Delta,\log n)$ rounds in the LOCAL model of distributed computing, where $n$ denotes the number of vertices of the input graph [J. Comb. Theory 2022]. Since then, the exponent in the $poly(\log n)$-part of the runtime has been improved by Christiansen [STOC 2023] and Bernshteyn and Dhawan [J. Comb. Theory, Series B, 2025]. However, the algorithms used in all of these works use large messages, leaving open the question for efficient algorithms in the more restrictive CONGEST model. We answer this question by presenting the first $poly(\Delta,\log n)$-round algorithm for $(\Delta + 1)$-edge coloring in the CONGEST model. Our algorithm is deterministic and the $n$-dependency of its runtime, $\tilde{O}(\log^5 n)$, matches the best published dependency in the LOCAL model.