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Register a new userOn Vizings edge colouring question
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Added by
Jonathan Narboni
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Soon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal edge colouring be reached from any given proper edge colouring through a series of Kempe changes? We answer this question in the affirmative for triangle-free graphs.
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Following a given ordering of the edges of a graph $G$, the greedy edge colouring procedure assigns to each edge the smallest available colour. The minimum number of colours thus involved is the chromatic index $chi(G)$, and the maximum is the so-called Grundy chromatic index. Here, we are interested in the restricted case where the ordering of the edges builds the graph in a connected fashion. Let $chi_c(G)$ be the minimum number of colours involved following such an ordering. We show that it is NP-hard to determine whether $chi_c(G)>chi(G)$. We prove that $chi(G)=chi_c(G)$ if $G$ is bipartite, and that $chi_c(G)leq 4$ if $G$ is subcubic.
A graph is apex if there is a vertex whose deletion makes the graph planar, and doublecross if it can be drawn in the plane with only two crossings, both incident with the infinite region in the natural sense. In 1966, Tutte conjectured that every two-edge-connected cubic graph with no Petersen graph minor is three-edge-colourable. With Neil Robertson, two of us showed that this is true in general if it is true for apex graphs and doublecross graphs. In another paper, two of us solved the apex case, but the doublecross case remained open. Here we solve the doublecross case; that is, we prove that every two-edge-connected doublecross cubic graph is three-edge-colourable. The proof method is a variant on the proof of the four-colour theorem.
Given $varepsilon>0$, there exists $f_0$ such that, if $f_0 le f le Delta^2+1$, then for any graph $G$ on $n$ vertices of maximum degree $Delta$ in which the neighbourhood of every vertex in $G$ spans at most $Delta^2/f$ edges, (i) an independent set of $G$ drawn uniformly at random has at least $(1/2-varepsilon)(n/Delta)log f$ vertices in expectation, and (ii) the fractional chromatic number of $G$ is at most $(2+varepsilon)Delta/log f$. These bounds cannot in general be improved by more than a factor $2$ asymptotically. One may view these as strong
This paper provides a survey of methods, results, and open problems on graph and hypergraph colourings, with a particular emphasis on semi-random `nibble methods. We also give a detailed sketch of some aspects of the recent proof of the ErdH{o}s-Faber-Lov{a}sz conjecture.
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