Vizing's Theorem in Near-Linear Time
<|reference_start|>Vizing's Theorem in Near-Linear Time: Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $\Delta$ can be \emph{edge colored} using at most $\Delta + 1$ 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 $\tilde O(mn^{1/3})$ by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to $\tilde O(mn^{1/4})$ time by [Bhattacharya, Costa, Solomon and Zhang, 2024]). We present an algorithm that computes a $(\Delta+1)$-edge coloring in $\tilde O(m)$ time -- in fact, even $O(m\log{\Delta})$ time -- with high probability, \emph{giving a near-optimal algorithm for this fundamental problem}.<|reference_end|>
@article{assadi2024vizing's,
    title={Vizing's Theorem in Near-Linear Time},
    author={Sepehr Assadi, Soheil Behnezhad, Sayan Bhattacharya, Mart'in Costa,
  Shay Solomon, Tianyi Zhang},
    journal={arXiv preprint arXiv:2410.05240},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.05240},
    primaryClass={cs.DS}
}