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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)
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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

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References

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