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On Vizing's edge colouring question
[article]

2021
*
arXiv
*
pre-print

Soon after his 1964 seminal paper

arXiv:2107.07900v1
fatcat:hvjcg6srrbc6pkvvtftnzlnpeu
*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. ...##
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On multiples of simple graphs and Vizing's Theorem

2010
*
Discrete Mathematics
*

So, the chromatic index χ of tG (the minimum number of

doi:10.1016/j.disc.2010.04.012
fatcat:gdpx4w2iqrg4vfuf6wghwzlod4
*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. ...##
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Achieving maximum chromatic index in multigraphs

2009
*
Discrete Mathematics
*

Let G be a multigraph with maximum degree ∆ and maximum

doi:10.1016/j.disc.2008.04.023
fatcat:zdgsn74dsnahnmb2ys5hfouqxq
*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. ...##
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A precolouring extension of Vizing's theorem
[article]

2018
*
arXiv
*
pre-print

If the

arXiv:1611.09002v2
fatcat:7lgybe3uejec3fbeutxdenkcxy
*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 . ...##
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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

1989
*
Journal of combinatorial theory. Series B (Print)
*

First

doi:10.1016/0095-8956(89)90045-2
fatcat:5vy3boy5hzdi7argnw5ipycuq4
*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. ...##
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Edge colouring by total labellings

2010
*
Discrete Mathematics
*

We introduce the concept of an

doi:10.1016/j.disc.2008.09.013
fatcat:4js6wv375veb3b5qvp5gyuaaei
*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 [20] . ...##
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A note on balanced colourings for lattice points

1990
*
Discrete Mathematics
*

It is not hard to see that the answer to the above

doi:10.1016/0012-365x(90)90227-9
fatcat:wway6hvrz5b37ltvyu3fxalco4
*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) . ...##
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Measurable versions of Vizing's theorem
[article]

2020
*
arXiv
*
pre-print

The "approximate" version states that, for any Borel probability measure

arXiv:1905.01716v2
fatcat:euuzel47hrhjhhfpvb3uq6mj3e
*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 [22] ). ...##
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Strong edge-colouring of sparse planar graphs
[article]

2014
*
arXiv
*
pre-print

In the last part of the paper, we raise some

arXiv:1401.4568v3
fatcat:me52jj5svnbgbpmmg36kg73s2a
*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 [5] ). ...##
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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*. ...##
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Strong edge-colouring of sparse planar graphs

2014
*
Discrete Applied Mathematics
*

In the last part of the paper, we raise some

doi:10.1016/j.dam.2014.07.006
fatcat:jyackqmk35hrllgg5tv3wpbooq
*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*). ...##
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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). ...##
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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 ...

##
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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

doi:10.47443/dml.2021.s105
fatcat:rcphxpoejvef5hty27uol6eegq
*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 [14] , 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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