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Rainbow Perfect Matchings in Complete Bipartite Graphs: Existence and Counting

Published online by Cambridge University Press:  09 July 2013

GUILLEM PERARNAU  and
ORIOL SERRA
GUILLEM PERARNAU
Affiliation:
Universitat Politècnica de Catalunya, Carrer Jordi Girona 1–3, 08034 Barcelona, Spain (e-mail: guillem.perarnau@ma4.upc.edu, oserra@ma4.upc.edu)
ORIOL SERRA
Affiliation:
Universitat Politècnica de Catalunya, Carrer Jordi Girona 1–3, 08034 Barcelona, Spain (e-mail: guillem.perarnau@ma4.upc.edu, oserra@ma4.upc.edu)
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Abstract

A perfect matching M in an edge-coloured complete bipartite graph Kn,n is rainbow if no pair of edges in M have the same colour. We obtain asymptotic enumeration results for the number of rainbow perfect matchings in terms of the maximum number of occurrences of each colour. We also consider two natural models of random edge-colourings of Kn,n and show that if the number of colours is at least n, then there is with high probability a rainbow perfect matching. This in particular shows that almost every square matrix of order n in which every entry appears n times has a Latin transversal.

Keywords

Primary 05A16Secondary 05B1505C1505D40

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 5 , September 2013 , pp. 783 - 799
Copyright
Copyright © Cambridge University Press 2013 

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