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Large Rainbow Matchings in Edge-Coloured Graphs

Published online by Cambridge University Press:  02 February 2012

Sobolev Institute of Mathematics, Novosibirsk 630090, Russia (e-mail: Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail:


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.

Copyright © Cambridge University Press 2012

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