Vizing's Theorem in Near-Linear Time

Abstract

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $\hat{I}”$ can be edge colored using at most $\hat{I}”+1$ different colors [Vizing, 1964]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $O(mn)$ time. This was subsequently improved to $\tilde{O}(m\sqrt{n})$ time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. Very recently, independently and concurrently, using randomization, this runtime bound was further improved to $\tilde{O}(n^2)$ by [Assadi, 2024] and $\tilde{O}(m{n}^{1/3})$ by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to $\tilde{O}(m{n}^{1/4})$ time by [Bhattacharya, Costa, Solomon and Zhang, 2024]). In this paper, we present a randomized algorithm that computes a $(\hat{I}”+1)$-edge coloring in near-linear time -- in fact, only $O(m\log \hat{I}”)$ time -- with high probability, giving a near-optimal algorithm for this fundamental problem.