Vizing's Theorem in Deterministic Almost-Linear Time

Abstract

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $\Delta$ can be edge colored using at most $\Delta + 1$ different colors. Vizing's original proof is easily translated into a deterministic $O(mn)$ time algorithm. This deterministic time bound was subsequently improved to $\tilde O(m \sqrt n)$ time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. A series of recent papers improved the time bound of $\tilde O(m\sqrt{n})$ using randomization, culminating in the randomized near-linear time $(\Delta+1)$-coloring algorithm by [Assadi, Behnezhad, Bhattacharya, Costa, Solomon, and Zhang, 2025]. At the heart of all of these recent improvements, there is some form of a sublinear time algorithm. Unfortunately, sublinear time algorithms as a whole almost always require randomization. This raises a natural question: can the deterministic time complexity of the problem be reduced below the $\tilde O(m\sqrt{n})$ barrier? In this paper, we answer this question in the affirmative. We present a deterministic almost-linear time $(\Delta+1)$-coloring algorithm, namely, an algorithm running in $m \cdot 2^{O(\sqrt{\log \Delta})} \cdot \log n = m^{1+o(1)}$ time. Our main technical contribution is to entirely forego sublinear time algorithms. We do so by presenting a new deterministic color-type sparsification approach that runs in almost-linear (instead of sublinear) time, but can be used to color a much larger set of edges.