The Power of Multi-Step Vizing Chains

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)