In this paper we show that properly edge-colored graphs G with |V(G)|<4δ(G)-3 have rainbow matchings of size δ(G); this gives the best known bound for a recent question of Wang. We also show that properly edge-colored graphs G with |V(G)|<2δ(G) have rainbow matchings of size at least δ(G)-2δ(G)2/3. This result extends (with a weaker error term) the well-known result that a factorization of the complete bipartite graph Kn,n has a rainbow matching of size n-o(n), or equivalently that every Latin square of order n has a partial transversal of size n-o(n) (an asymptotic version of the Ryser-Brualdi conjecture). In this direction we also show that every Latin square of order n has a cycle-free partial transversal of size n-o(n).
- Partial transversals in Latin squares
- Proper edge colorings
- Rainbow matchings
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics