Vizing's Theorem in Near-Linear Time

Vizing's theorem states that any.. -vertex..-edge graph of maximum degree. can be edge colored using at most. + 1 different colors [Vizing, 1964]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in.. (....) time. This was subsequently improved to 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 by [Assadi, 2024] and by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to by [Bhattacharya, Costa, Solomon and Zhang, 2024]). In this paper, we present a randomized algorithm that computes a (.+ 1)-edge coloring in near-linear time-in fact, only.. (.. log.) time-with high probability, giving a near-optimal algorithm for this fundamental problem.