Large rainbow matchings in edge-coloured graphs

  • Authors:

  • Alexandr Kostochka;Matthew Yancey

  • Affiliations:

  • Sobolev institute of mathematics, novosibirsk 630090, russia (e-mail: kostochk@math.uiuc.edu) and department of mathematics, university of illinois, urbana, il 61801, usa (e-mail: yancey1@illinois ...;Department of mathematics, university of illinois, urbana, il 61801, usa (e-mail: yancey1@illinois.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2012

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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 â聣楼 4, every edge-coloured graph with minimum colour degree k contains a rainbow matching of size at least â聦聢k/2â聦聣. A properly edge-coloured K4 has no such matching, which motivates the restriction k â聣楼 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 â聦聤k/2â聦聥 is guaranteed to exist, and they proved several sufficient conditions for a matching of size â聦聢k/2â聦聣. We prove the conjecture in full.