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^{7}\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：近期论文[Ber'2022]、[GP'2020]、[DHZ'2019]通过连接或拼接多个Vizing链，形成了Bernshteyn[Ber'2022]所称的“多步Vizing链”，从而解决了(Δ+1)-边染色问题的不同变体。在本文中，我们对该术语提出了一个稍广义的定义。随后，我们应用多步Vizing链构造来证明边染色的组合性质，从而在不同计算模型下推导出（改进的）边染色算法。这种方法在构造尊重某种局部性的增广子图时尤为有效。首先，我们构造了严格局部多步Vizing链，并用它们证明Vizing定理的局部版本，从而证实了Bonamy、Delcourt、Lang和Postle[BDLP'2020]近期的一个猜想。我们的证明是构造性的，并隐含了计算此类染色的算法。然后，我们证明对于任何未染色边，存在大小为O(Δ^{7} log n)的增广子图，回答了Bernshteyn[Ber'2022]的一个开放问题。Chang、He、Li、Pettie和Uitto[CHLPU'2018]证明了此类增广子图大小的下界为Ω(Δ log n/Δ)，因此该上界在Δ和常数因子意义下是紧的。这些思想还可扩展到更快的确定性LOCAL算法，用于(Δ+1)-边染色，其运行时间为Õ(poly(Δ) log^6 n)轮。这些结果改进了Bernshteyn[Ber'2022]的近期突破性成果，后者证明了大小为O(Δ^6 log^2 n)的增广子图的存在性，并由此给出了LOCAL模型中首个运行时间为O(poly(Δ, log n))轮的(Δ+1)-边染色算法……（参见论文获取摘要其余部分）