Edge-coloring sparse graphs with $\Delta$ colors in quasilinear time

Abstract

In this paper we show that every graph $G$ of bounded maximum average degree ${\rm mad}(G)$ and with maximum degree $\Delta$ can be edge-colored using the optimal number of $\Delta$ colors in quasilinear time, whenever $\Delta\ge 2{\rm mad}(G)$. The maximum average degree is within a multiplicative constant of other popular graph sparsity parameters like arboricity, degeneracy or maximum density. Our algorithm extends previous results of Chrobak and Nishizeki [J. Algorithms, 1990] and Bhattacharya, Costa, Panski and Solomon [ESA 2024].