  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)
