Chromatic index of dense quasirandom graphs

Let G be a simple graph with maximum degree ∆(G). A subgraph H of G is overfull if |E(H)| > ∆(G)⌊|V (H)|/2⌋. Chetwynd and Hilton in 1985 conjectured that a graph G on n vertices with ∆(G) > n/3 has chromatic index ∆(G) if and only if G contains no overfull subgraph. Glock, Kühn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. In this paper, we show that the conjecture of Glock, Kühn and Osthus is affirmative.

[1]  P. D. Seymour,et al.  On Multi‐Colourings of Cubic Graphs, and Conjectures of Fulkerson and Tutte , 1979 .

[2]  G. Nemhauser,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2014 .

[3]  C. McDiarmid,et al.  The Solution of a Timetabling Problem , 1972 .

[4]  Daniela Kühn,et al.  Optimal path and cycle decompositions of dense quasirandom graphs , 2015, Electron. Notes Discret. Math..

[5]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[6]  A. Hilton,et al.  Star multigraphs with three vertices of maximum degree , 1986 .

[7]  D. König.  Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre , 1916 .

[8]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[9]  Jayadev Misra,et al.  A Constructive Proof of Vizing's Theorem , 1992, Inf. Process. Lett..

[10]  Anthony J. W. Hilton,et al.  The Edge-Chromatic Class of Graphs with Maximum Degree at Least |V| – 3 , 1988 .

[11]  Robin J. Wilson,et al.  Edge-colourings of graphs , 1977 .

[12]  D. Kuhn,et al.  Proof of the 1-factorization and Hamilton Decomposition Conjectures , 2014, 1401.4183.

[13]  S. Hakimi.  On Realizability of a Set of Integers as Degrees of the Vertices of a Linear Graph. I , 1962 .

[14]  Michael Plantholt,et al.  Overfull conjecture for graphs with high minimum degree , 2004, J. Graph Theory.

[15]  Ian Holyer,et al.  The NP-Completeness of Edge-Coloring , 1981, SIAM J. Comput..

[16]  V. G. Vizing.  The chromatic class of a multigraph , 1965 .

[18]  Emil R. Vaughan,et al.  An Asymptotic Version of the Multigraph 1‐Factorization Conjecture , 2010, J. Graph Theory.

[19]  Ding‐Zhu Du,et al.  Wiley Series in Discrete Mathematics and Optimization , 2014 .

[20]  Eckhard Steffen,et al.  Independent sets and 2-factors in edge-chromatic-critical graphs , 2004, J. Graph Theory.