Recent papers [Ber'2022], [GP'2020], [DHZ'2019] have addressed different variants of the (\Delta + 1)-edge colouring problem by concatenating or gluing together many Vizing chains to form what Bernshteyn [Ber'2022] coined \emph{multi-step Vizing chains}. In this paper, we propose a slightly more general definition of this term. We then apply multi-step Vizing chain constructions to prove combinatorial properties of edge colourings that lead to (improved) algorithms for computing edge colouring across different models of computation. This approach seems especially powerful for constructing augmenting subgraphs which respect some notion of locality. First, we construct strictly local multi-step Vizing chains and use them to show a local version of Vizings Theorem thus confirming a recent conjecture of Bonamy, Delcourt, Lang and Postle [BDLP'2020]. Our proof is constructive and also implies an algorithm for computing such a colouring. Then, we show that for any uncoloured edge there exists an augmenting subgraph of size O(\Delta^{6}\log n), answering an open problem of Bernshteyn [Ber'2022]. Chang, He, Li, Pettie and Uitto [CHLPU'2018] show a lower bound of \Omega(\Delta \log \frac{n}{\Delta}) for the size of such augmenting subgraphs, so the upper bound is tight up to \Delta and constant factors. These ideas also extend to give a faster deterministic LOCAL algorithm for (\Delta + 1)-edge colouring running in \tilde{O}(\poly(\Delta)\log^6 n) rounds. These results improve the recent breakthrough result of Bernshteyn [Ber'2022], who showed the existence of augmenting subgraphs of size O(\Delta^6\log^2 n), and used these to give the first (\Delta + 1)-edge colouring algorithm in the LOCAL model running in O(\poly(\Delta, \log n)) rounds. ... (see paper for the remaining part of the abstract)
翻译:最近的论文 [Ber'2022, [NGDD2020], [DHZ2019] 已经解决了不同变式的( Delta + 1) 前沿色化问题, 将许多 Vizing 链条混为一体, 形成 Bernshteyn [Ber'2022] 调出 mulph{Lemph{多步Vizing 链 。 在本文中, 我们建议略微笼统地定义此术语。 然后我们应用多步 Visization 链构造来证明边缘色化的组合性特性, 导致( 简化) 在不同计算模型中计算边际色化的( Delta + 1) 色化。 这个方法似乎特别强大, 构建尊重某些地点概念的扩大子色化链 。 首先, 我们构建了严格的本地多步的 Vishetrical Vizem 模式, 从而确认了 Bonamiy、 Delcourt、 Lang 和 Postle 的元化 。 我们的证据是具有建设性性的, 也意味着在计算这种颜色大小的亚化。 然后, 我们展示了一个不连续的奥氏 直端端 。