Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-14T08:31:16.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Large Rainbow Matchings in Edge-Coloured Graphs

Published online by Cambridge University Press:  02 February 2012

ALEXANDR KOSTOCHKA  and
MATTHEW YANCEY
ALEXANDR KOSTOCHKA
Affiliation:
Sobolev Institute of Mathematics, Novosibirsk 630090, Russia (e-mail: kostochk@math.uiuc.edu) Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: yancey1@illinois.edu)
MATTHEW YANCEY
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: yancey1@illinois.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

A rainbow subgraph of an edge-coloured graph is a subgraph whose edges have distinct colours. The colour degree of a vertex v is the number of different colours on edges incident with v. Wang and Li conjectured that for k ≥ 4, every edge-coloured graph with minimum colour degree k contains a rainbow matching of size at least ⌈k/2⌉. A properly edge-coloured K4 has no such matching, which motivates the restriction k ≥ 4, but Li and Xu proved the conjecture for all other properly coloured complete graphs. LeSaulnier, Stocker, Wenger and West showed that a rainbow matching of size ⌊k/2⌋ is guaranteed to exist, and they proved several sufficient conditions for a matching of size ⌈k/2⌉. We prove the conjecture in full.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 1-2: Honouring the Memory of Richard H. Schelp , March 2012 , pp. 255 - 263
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Kano, M. and Li, X. (2008) Monochromatic and heterochromatic subgraphs in edge-colored graphs: A survey. Graphs Combin. 24 237–263.Google Scholar
[2]LeSaulnier, T. D., Stocker, C., Wegner, P. S. and West, D. B. (2010) Rainbow matching in edge-colored graphs. Electron. J. Combin. 17 N26.CrossRefGoogle Scholar
[3]Li, H. and Wang, G. (2008) Color degree and heterochromatic matchings in edge-colored bipartite graphs. Util. Math. 77 145–154.Google Scholar
[4]Li, X. and Xu, Z. (2007) On the existence of a rainbow 1-factor in proper coloring of Krn (r). arXiv:0711.2847 [math.CO].Google Scholar
[5]Wang, G. and Li, H. (2008) Heterochromatic matchings in edge-colored graphs. Electron. J. Combin. 15 R138.CrossRefGoogle Scholar