Rainbow matchings and cycle-free partial transversals of Latin squares

Abstract

In this paper we consider properly edge-colored graphs, i.e. two edges with the same color cannot share an endpoint, so each color class is a matching.

On the number of symbols that forces a transversal

“…Thus we obtain a larger matching, but M 2 may not be rainbow, due to repeating the colours of at least one of a 0 b and ab 0 . While the current matching M 2 is not rainbow, we apply the following trace back algorithm (similar to that of [6]). Algorithm 4.5.…”

Rainbow matchings in strongly edge-colored graphs

“…He pointed out that since a properly edge-colored K n,n for an even number n does not always have a perfect rainbow matching, the lowest possible value would be 2δ(G). Several positive answers were obtained for this question, and Gyárfás et al [3] showed that every properly edge-colored graph G with |V (G)| ≥ 4δ(G) − 3 contains a rainbow matching of size δ(G). A result by Lo and Tan [7] for arbitrary edge-colored graphs implies that if δ(G) ≥ 4, then any properly edgecolored graph G with |V (G)| ≥ 4δ(G) − 4 contains a rainbow matching of size δ(G).…”

“…There have been several studies giving lower bounds for the size of a maximum rainbow matching in a properly edgecolored graph G in terms of its minimum degree δ(G) [10,6,3]. Wang [10] asked whether it is possible to obtain a function f such that any properly edge-colored graph G having more than f (δ(G)) vertices is guaranteed to contain a rainbow matching of size δ(G).…”

“…Gyárfás et al [3] showed that if G is a properly edge-colored graph with |V (G)| ≥ 2δ(G), then G contains a rainbow matching of size at least δ − 2δ(G) 2 3 . Since lower bounds for the size of maximum rainbow matchings in properly edge-colored graphs are well studied, it is natural to try to improve the lower bounds under stronger assumptions on the edge-coloring.…”

Large matchings in bipartite graphs have a rainbow matching

“…During the last decade the problem of finding conditions for large rainbow matchings in a graph was extensively explored. See, for example, [1,8,10,11]. Many results and conjectures on the subject were influenced by the well-known conjectures of Ryser [12], asserting that every Latin square of odd order n has a transversal of order n, and Brualdi [6] (see also [5] p. 255), asserting that every Latin square of even order n has a partial transversal of size n − 1.…”