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On Vizing's theorem, adjacency lemma and fan argument generalized to multigraphs

K.H. Chew
<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
From this result, we show that the multigraph version of Vizing's theorem, Vizing's adjacency lemma and Vizing's fan argument can be obtained immediately.  ...  The proof given here is based on fan and counting arguments involving small number of colour changes and is considerably simpler and shorter than the one in Ehrenfeucht et al. (1984) .  ...  If in addition, M is simple and critical, that is, M is connected and every edge of M is critical, then Theorem 4 is true for every pair of two adjacent vertices x and y and this is then the familar Vizing's  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(96)00086-6">doi:10.1016/s0012-365x(96)00086-6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ul62fzpf6refjapej7owdxazme">fatcat:ul62fzpf6refjapej7owdxazme</a> </span>
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Page 44 of Mathematical Reviews Vol. , Issue 98A [page]

<span title="">1998</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
|Chew, Kian Hoe] (5-NSW-SM; Sydney) On Vizing’s theorem, adjacency lemma and fan argument generalized to multigraphs. (English summary) Discrete Math. 171 (1997), no. 1-3, 283-286.  ...  From this result, we show that the multigraph version of Vizing’s theorem, Vizing’s adjacency lemma and Vizing’s fan argument can be obtained immediately.  ... 
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Author index to volume 171 (1997)

<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Spicer, Embeddings of m-cycle systems and incomplete m-cycle systems: m ~< 14 (1 3) 55 75 Chew, K.H., On Vizing's theorem, Adjacency lemma and fan argument generalized to multigraphs (Note) (I -3) 283-  ...  Haile, Remarks on the size of critical edge-chromatic graphs (Note) (1 3) 287 293 da Silva, I.P.F., Note on inseparability graphs of matroids having exactly one class of orientations (1-3) 77 87 Dragan  ... 
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A brief history of edge-colorings – with personal reminiscences

<span title="2021-03-11">2021</span> <i title="Shahin Digital Publisher"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bdwdrpk3ire47o4q6hmknhdqya" style="color: black;">Discrete Mathematics Letters</a> </i> &nbsp;
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.  ...  we should look at Vizing's and Kierstead's proofs, based on different trees (fans and paths), and try to find a common generalization.  ...  Vizing's adjacency lemma has proved to be extremely useful in a number of contexts.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.47443/dml.2021.s105">doi:10.47443/dml.2021.s105</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rcphxpoejvef5hty27uol6eegq">fatcat:rcphxpoejvef5hty27uol6eegq</a> </span>
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Edge-coloring via fixable subgraphs [article]

Daniel W. Cranston, Landon Rabern
<span title="2015-07-20">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
at least one such configuration; these configurations are called reducible for that theorem.  ...  Many graph coloring proofs proceed by showing that a minimal counterexample to the theorem being proved cannot contain certain configurations, and then showing that each graph under consideration contains  ...  This result immediately implies the fan equation, which is an extension of Vizing's Adjacency Lemma to multigraphs and a standard tool in proving reducibility for edge-coloring (see [9, p. 19ff] ).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1507.05600v1">arXiv:1507.05600v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7dq6nwohnngjpgybo5gj3gqacy">fatcat:7dq6nwohnngjpgybo5gj3gqacy</a> </span>
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A local strengthening of Reed's ω, Δ, χ conjecture for quasi-line graphs [article]

Maria Chudnovsky, Andrew D. King, Matthieu Plumettaz, Paul Seymour
<span title="2011-11-29">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Our proofs lead to polytime algorithms for constructing colourings that achieve our bounds: $O(n^2)$ for line graphs and $O(n^3m^2)$ for quasi-line graphs.  ...  Reed's $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph satisfies $\chi\leq \lceil\frac 12(\Delta+1+\omega)\rceil$; it is known to hold for all claw-free graphs.  ...  These fans generalize Vizing's fans, originally used in the proof of Vizing's theorem [13] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1109.2112v2">arXiv:1109.2112v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5uhrlkrffnbkjf2wr5gooxaumy">fatcat:5uhrlkrffnbkjf2wr5gooxaumy</a> </span>
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A Local Strengthening of Reed's $\omega$, $\Delta$, $\chi$ Conjecture for Quasi-line Graphs

Maria Chudnovsky, Andrew D. King, Matthieu Plumettaz, Paul Seymour
<span title="">2013</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hn6wut4eynat5my6xdfsebg7fm" style="color: black;">SIAM Journal on Discrete Mathematics</a> </i> &nbsp;
Our proofs lead to polytime algorithms for constructing colourings that achieve our bounds: O(n 2 ) for line graphs and O(n 3 m 2 ) for quasi-line graphs.  ...  Reed's ω, ∆, χ conjecture proposes that every graph satisfies χ ≤ 1 2 (∆ + 1 + ω) ; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture.  ...  These fans generalize Vizing's fans, originally used in the proof of Vizing's theorem [12] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/110847585">doi:10.1137/110847585</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/x5kwchj7bfestan3iyki2qhobi">fatcat:x5kwchj7bfestan3iyki2qhobi</a> </span>
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A superlocal version of Reed's Conjecture [article]

Katherine Edwards, Andrew D. King
<span title="2014-11-15">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We provide some fundamental evidence in support, namely that the stronger bound holds in the fractional relaxation and holds for both quasi-line graphs and graphs with stability number two.  ...  the idea that the chromatic number cannot be greatly affected by any particular stable set of vertices, we propose a further strengthening that considers a bound supplied by the neighbourhoods of two adjacent  ...  Acknowledgements We thank the referee for a careful and helpful review, and the editors for their time and contribution to the journal.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1208.5188v2">arXiv:1208.5188v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qnhkgk4jv5dnheiv53i7vj6fza">fatcat:qnhkgk4jv5dnheiv53i7vj6fza</a> </span>
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On Vizing's edge colouring question [article]

Marthe Bonamy, Oscar Defrain, Tereza Klimošová, Aurélie Lagoutte, Jonathan Narboni
<span title="2021-07-16">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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 thank Caroline Brosse, Vincent Limouzy, Carole Muller and Lucas Pastor for extensive discussions around this topic.  ...  We gratefully acknowledge support from Nicolas Bonichon and the Simon family for the organisation of the 5 th Pessac Graph Workshop, where a preliminary part of this research was done.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2107.07900v1">arXiv:2107.07900v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hvjcg6srrbc6pkvvtftnzlnpeu">fatcat:hvjcg6srrbc6pkvvtftnzlnpeu</a> </span>
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Extension from Precoloured Sets of Edges [article]

Katherine Edwards, António Girão, Jan van den Heuvel, Ross J. Kang, Gregory J. Puleo, Jean-Sébastien Sereni
<span title="2018-05-27">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi  ...  This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability.  ...  Acknowledgement The authors would like to thank the anonymous referees for their meticulous reading and for their corrections and suggestions, which improved the article significantly.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1407.4339v3">arXiv:1407.4339v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2dcvaq7c6var5gx657gymstk74">fatcat:2dcvaq7c6var5gx657gymstk74</a> </span>
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Extension from Precoloured Sets of Edges

Katherine Edwards, António Girão, Jan Van den Heuvel, Ross J. Kang, Gregory J. Puleo, Jean-Sébastien Sereni
<span title="2018-07-13">2018</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi  ...  This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability.  ...  Acknowledgement The authors would like to thank the anonymous referees for their meticulous reading and for their corrections and suggestions, which improved the article significantly.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/6303">doi:10.37236/6303</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cxjggaybn5gshdbesmnq4e6tky">fatcat:cxjggaybn5gshdbesmnq4e6tky</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200217035734/https://www.combinatorics.org/ojs/index.php/eljc/article/download/v25i3p1/pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/71/3a/713a56156bf526cff5fa693522b8ee690da8ed5d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/6303"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Edge-Coloring Partialk-Trees

Xiao Zhou, Shin-ichi Nakano, Takao Nishizeki
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/anp7kqfyerdljfxydqb4objj6a" style="color: black;">Journal of Algorithms</a> </i> &nbsp;
The edge-coloring problem is one of the well-known combinatorial problems for which no efficient algorithms were previously known, except a polynomial-time algorithm of very high complexity.  ...  This paper gives a linear-time sequential algorithm and an optimal parallel algorithm which find an edge-coloring of a given partial k-tree with the minimum number of colors for fixed k. ᮊ  ...  . ⌬ The following lemma is an expression of a classical result on ''critical w x graphs,'' called ''Vizing's adjacency lemma' ' 11, 19, 22 . Ž . LEMMA 3.1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jagm.1996.0061">doi:10.1006/jagm.1996.0061</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/d2hvbbucnnggvmwh2dzl3ksxyi">fatcat:d2hvbbucnnggvmwh2dzl3ksxyi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170830034900/http://www.ecei.tohoku.ac.jp/alg/nishizeki/sub/j/DVD/PDF_J/J106.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d0/88/d0888ef9d3a5778b8a0cdb5e003104a156adbc0b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jagm.1996.0061"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Average degrees of edge-chromatic critical graphs [article]

Yan Cao, Guantao Chen, Suyun Jiang, Huiqing Liu, Fuliang Lu
<span title="2017-08-03">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Additionally, Woodall constructed an infinite family of graphs showing his result cannot be improved by well-known Vizing's Adjacency Lemma and other known edge-coloring techniques.  ...  To over come the barrier, we follow the recently developed recoloring technique of Tashkinov trees to expand Vizing fans technique to a larger class of trees.  ...  In the same paper, Woodall provided the following example demonstrating that the above result cannot be improved by the use of his new adjacency Lemmas (see Lemma 2 and Lemma 3) and Vizing's Adjacency  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1708.01279v1">arXiv:1708.01279v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tl53zl47tzgypdfo4khyycekxa">fatcat:tl53zl47tzgypdfo4khyycekxa</a> </span>
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Master index of volumes 171–180

<span title="">1998</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Rhodes, Avoiding partial Latin squares and intricacy Chew, K.H., On Vizing's theorem, Adjacency lemma and fan argument generalized to multigraphs (Note) Chia, G.L., A bibliography on chromatic polynomials  ...  ., On the Oberwolfach problem for complete multigraphs Gyarfas, A., Z. Kiraly and J. Lehel, On-line 3R., see M.O.  ... 
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Approximating the chromatic index of multigraphs

Guantao Chen, Xingxing Yu, Wenan Zang
<span title="2009-05-29">2009</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/tsm6lnbvf5ahzjp7czojbjig5i" style="color: black;">Journal of combinatorial optimization</a> </i> &nbsp;
It is well known that if G is a multigraph then χ (G) ≥ χ  ...  Recall Vizing's "fan sequence" (or "star") argument for proving that χ (G) ∈ {∆(G), ∆(G) + 1} when G is a simple graph.  ...  what Vizing's theorem does for simple graphs.  ... 
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