Abstract
Aharoni and Berger conjectured that every collection of $n$ matchings of size $n+1$ in a bipartite graph contains a rainbow matching of size $n$. This conjecture is related to 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. The conjecture is known to hold when the matchings are much larger than $n+1$. The best bound is currently due to Aharoni, Kotlar, and Ziv who proved the conjecture when the matchings are of size at least $3n/2+1$. When the matchings are all edge-disjoint and perfect, 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 $\phi n+o(n)$ where $\phi\approx 1.618$ 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.
Highlights
A Latin square of order n is an n × n array filled with n different symbols, where no symbol appears in the same row or column more than once
The electronic journal of combinatorics 22 (2015), #P00 branches of mathematics such as algebra and experimental design. They occur in recreational mathematics—for example completed Sudoku puzzles are Latin squares
One reason transversals in Latin squares are interesting is that a Latin square has an orthogonal mate if, and only if, it has a decomposition into disjoint transversals
Summary
A Latin square of order n is an n × n array filled with n different symbols, where no symbol appears in the same row or column more than once. Gyarfas, and Sarkozy proved something a bit more general in [4]—for every k, they gave an upper bound on the number of colours needed to find a rainbow matching of size n − k Another approximate version of Conjecture 1.3 comes from Theorem 1.2. G can be decomposed into disjoint rainbow matchings of size n” (to see that this is equivalent to Theorem 1.2, associate an m-edge coloured bipartite graph with any m×n Latin rectangle by placing a colour k edge between i and j whenever (k, i) has symbol j in the rectangle). The key intermediate result we prove is that every properly edge coloured directed graph D has a rainbow k-edge-connected subset C of size roughly δ+(D). If the various parameters are chosen suitably, we obtain that the original matching M must have used every colour i.e. Theorem 1.5
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