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### On Vizing's edge colouring question [article]

Marthe Bonamy, Oscar Defrain, Tereza Klimošová, Aurélie Lagoutte, Jonathan Narboni
2021 arXiv   pre-print
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.  ...  Observe that a Kempe chain may consist in a single edge e, coloured say α, when some colour β does not appears on any edge incident to e.  ...

### On multiples of simple graphs and Vizing's Theorem

J.M. McDonald
2010 Discrete Mathematics
So, the chromatic index χ of tG (the minimum number of colours needed to colour the edges of tG such that adjacent edges receive different colours), is at least td.  ...  Let G be a simple connected graph with maximum degree d, and let tG denote the graph obtained from G by replacing each edge with t parallel edges. Vizing's Theorem says that td ≤ χ (tG) ≤ td + t.  ...  Let G be the simple graph with d + 2 vertices obtained from a 1-regular graph with d + 1 vertices by adding one edge incident to the remaining vertex.  ...

### Achieving maximum chromatic index in multigraphs

J.M. McDonald
2009 Discrete Mathematics
Let G be a multigraph with maximum degree ∆ and maximum edge multiplicity µ. Vizing's Theorem says that the chromatic index of G is at most ∆ + µ.  ...  We prove that, with the exception of µK 3 , every connected G with µ ≥ 2 which achieves Vizing's upper bound must contain a specific dense subgraph on five vertices.  ...  Moreover, since T is maximal, all these colours must be on edges induced by V(T). So each vertex of T must see at least nµ − (n − 1) colours on edges which do not leave the tree.  ...

### A precolouring extension of Vizing's theorem [article]

António Girão, Ross J. Kang
2018 arXiv   pre-print
If the edges of M are arbitrarily precoloured from K, then there is guaranteed to be a proper edge-colouring using only colours from K that extends the precolouring on M to the entire graph.  ...  Fix a palette K of Δ+1 colours, a graph with maximum degree Δ, and a subset M of the edge set with minimum distance between edges at least 9.  ...  . , ∆ + µ}, then there is a proper edge-colouring of G using colours from K that agrees with the precolouring on M .  ...

### Page 1676 of Mathematical Reviews Vol. , Issue 2000c [page]

2000 Mathematical Reviews
This remarkable result is extended here to multigraphs whose underlying simple graph becomes bipartite after removing one edge (this question was suggested by Woodall).  ...  We color each undirected edge in one color and each directed edge in two colors, such that the color of the first half of a directed edge is smaller than the color of the second half.  ...

### On the Δ-subgraph of graphs which are critical with respect to the chromatic index

A.G Chetwynd, A.J.W Hilton, D.G Hoffman
1989 Journal of combinatorial theory. Series B (Print)
First colour the n + 2 edges on B* with the colours ci, . . . . c,,~, using each colour exactly once.  ...  Let e, = b,bl be an edge within B adjacent to e. Proceed as before to colour G\e,. Let ci be the colour on the edge of the extended graph joining b* and bl.  ...

### Edge colouring by total labellings

Stephan Brandt, Kristína Budajová, Dieter Rautenbach, Michael Stiebitz
2010 Discrete Mathematics
We introduce the concept of an edge-colouring total k-labelling.  ...  This is a labelling of the vertices and the edges of a graph G with labels 1, 2, . . . , k such that the weights of the edges define a proper edge colouring of G.  ...  Let c : E(G) → {1, 2, . . . , ∆ + 1} be a proper edge colouring of G which exists by Vizing's Theorem  .  ...

### A note on balanced colourings for lattice points

Jin Akiyama, Jorge Urrutia
1990 Discrete Mathematics
It is not hard to see that the answer to the above question is "yes". In this note we generalize this result, and show that P n can be coloured with m (m≥2) colours in such  ...  Is it always possible to colour some of the points red and the remaining points white in such a way that, for any straight line L parallel to either one of the coordinate axes, the difference (in absolute  ...  Thus by Vizing's Theorem, H is m-edge colourable. However any such colouring of H induces (by using ƒ) an m-colouring of P n . (See Figure 2a,2b) .  ...

### Measurable versions of Vizing's theorem [article]

Jan Grebík, Oleg Pikhurko
2020 arXiv   pre-print
The "approximate" version states that, for any Borel probability measure on the edge set and any ϵ>0, we can properly colour all but ϵ-fraction of edges with Δ+π colours in a Borel way.  ...  We establish two versions of Vizing's theorem for Borel multi-graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively Δ and π.  ...  Marks [22, Question 4.9] asked if a measurable version of Vizing's theorem holds for arbitrary Borel probability measures: Question 1.5 (Marks  ).  ...

### Strong edge-colouring of sparse planar graphs [article]

Julien Bensmail, Hervé Hocquard
2014 arXiv   pre-print
In the last part of the paper, we raise some questions related to a long-standing conjecture of Vizing on proper edge-colouring of planar graphs.  ...  A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching.  ...  Perhaps the most challenging question for strong edge-colouring is the following conjecture: Conjecture 1 (Erdős and Nešetřil  ).  ...

### Page 6042 of Mathematical Reviews Vol. , Issue 2000i [page]

2000 Mathematical Reviews
This answers one question of X. D. Zhu [J. Graph Theory 16 (1992), no. 6, 557-569; MR 93i:05066].  ...  Improving the known Vizing’s bound it is shown that 7'(G) < A(G) + [,/a(G)] for any multigraph G in which every cycle of length greater than 2 contains a simple edge.  ...

### Strong edge-colouring of sparse planar graphs

Julien Bensmail, Ararat Harutyunyan, Hervé Hocquard, Petru Valicov
2014 Discrete Applied Mathematics
In the last part of the paper, we raise some questions related to a long-standing conjecture of Vizing on proper edge-colouring of planar graphs.  ...  A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching.  ...  we colour uu k−3 , vv k−3 and u k−3 v k−3 in this order (at each step we have at least one colour left for the current edge).  ...

### Page 8940 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews
We focus our interest on graph classes (defined in terms of forbidden induced sub- graphs) for which the question of 3-colourability can be decided in polynomial time and, if so, a proper 3-colouring can  ...  On the positive side, the author establishes APX results when R consists of paths, all of whose edges belong to one row and one column of the mesh or torus (Ic-paths).  ...

### Page 2217 of Mathematical Reviews Vol. , Issue 2001D [page]

2001 Mathematical Reviews
Vizing’s well-known result says that if the maximum degree of a graph G is A, then G is either (A + 1)-edge-colorable or A-edge- colorable. In the latter case G is said to be in class one.  ...  —3; (iii) any graph with A > 11 embeddable in a surface of characteristic —4 or =- The authors finally pose some questions regarding edge colorings of graphs embeddable on surfaces and the Heawood number  ...

### A brief history of edge-colorings – with personal reminiscences

2021 Discrete Mathematics Letters
In this article we survey some important milestones in the history of edge-colorings of graphs, from the earliest contributions of Peter Guthrie Tait and Dénes König to very recent work.  ...  Shortly afterwards, Stanley Fiorini and I collaborated on the book, Edge-colourings of graphs  , which was based on his thesis and appeared in 1977.  ...  To understand the importance of Tashkinov trees, we return to Vizing's proof that, for a multigraph G, χ (G) ≤ ∆(G) + µ(G); it was based on the use of Kempe chains to recolor the edges of multi-fans.  ...
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