Work

Vizing’s Theorem in Near-Linear Time

Year: 2026

Type: article

Abstract: Vizing’s theorem states that any n -vertex m -edge graph of maximum degree Δ can be edge colored using at most Δ + 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]. 1 Very recently, independently and concurrently, using randomization, this runtime bound was further improved to \(\tilde{O}(n^2)\) by [Assadi, 2024] and \(\tilde{O}(mn^{1/3})\) by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to \(\tilde{O}(mn^{1/4})\) by [Bhattacharya, Costa, Solomon and Zhang, 2024]). In this article, we present a randomized algorithm that computes a Δ + 1-edge coloring in near-linear time—in fact, only O(m log Δ) time—with high probability, giving a near-optimal algorithm for this fundamental problem .

Source: Journal of the ACM

DOI: 10.1145/3806392

Authors: Sepehr Assadi, Soheil Behnezhad, Sayan Bhattacharya, Martin Costa, Shay Solomon, Tianyi Zhang