Even Faster (∆ + 1)-Edge Coloring via Shorter Multi-Step Vizing Chains

Vizing’s Theorem from 1964 states that any n-vertex m-edge graph with maximum degree ∆ can be edge colored using at most ∆ + 1 colors. For over 40 years, the state-of-the-art running time for computing such a coloring, obtained independently by Arjomandi [1982] and by Gabow, Nishizeki, Kariv, Leven and Terada [1985], was O͂(m√n). Very recently, this time bound was improved in two independent works, by Bhattacharya, Carmon, Costa, Solomon and Zhang to O͂(mn¹/³), and by Assadi to O͂(n²). In this paper we present an algorithm that computes such a coloring in O͂(mn¹/⁴) time. Our key technical contribution is a subroutine for extending the coloring to one more edge within time O͂(∆² + ^√∆n). The best previous time bound of any color extension subroutine is either the trivial O(n), dominated by the length of a Vizing chain, or the bound O͂(∆⁶) by Bernshteyn [2022], dominated by the length of multi-step Vizing chains, which is basically a concatenation of multiple (carefully chosen) Vizing chains. Our color extension subroutine produces significantly shorter multi-step Vizing chains than in previous works, for sufficiently large ∆.