Rainbow Perfect Matchings in Complete Bipartite Graphs: Existence and Counting

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 nat...

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Bibliographic Details
Published inCombinatorics, probability & computing Vol. 22; no. 5; pp. 783 - 799
Main Authors PERARNAU, GUILLEM, SERRA, ORIOL
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.09.2013
Subjects
Asymptotic properties
Color
Colour
Combinatorial analysis
Counting
Graphs
Matching
Rainbows
Secondary 05B15
Primary 05A16
05D40
05C15
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Summary: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.
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ISSN:0963-5483
1469-2163
DOI:10.1017/S096354831300028X