Rainbow matchings and rainbow connectedness

Pokrovskiy, Alexey (2017) Rainbow matchings and rainbow connectedness. Electronic Journal of Combinatorics 24 (1), pp. 1-13. ISSN 1077-8926.

Abstract

Aharoni and Berger conjectured that every bipartite graph which is the union of n matchings of size n + 1 contains a rainbow matching of size n. This conjecture is a generalization of several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. There have been many recent partial results about the Aharoni-Berger Conjecture. In the case when the matchings are much larger than n + 1, the best bound is currently due to Clemens and Ehrenmüller who proved the conjecture when the matchings are of size at least 3n/2 + o(n). When the matchings are all edge-disjoint and perfect, then the best result follows from a theorem of Häggkvist and Johansson which implies the conjecture when the matchings have size at least n + o(n). In this paper we show that the conjecture is true when the matchings have size n + o(n) and are all edge-disjoint (but not necessarily perfect). We also give an alternative argument to prove the conjecture when the matchings have size at least φn+o(n) where φ≈1.618 is the Golden Ratio. Our proofs involve studying connectedness in coloured, directed graphs. The notion of connectedness that we introduce is new, and perhaps of independent interest.

Metadata

Statistics

DownloadsShow export options

Export as XMLJSONCSV

Activity Overview

6 month trend

168Downloads

6 month trend

135Hits

Additional statistics are available via IRStats2.

Archive Staff Only (login required)

Edit/View Item