Leírás
The well-known Ryser-Brualdi-Stein Conjecture states that any proper edge-coloring of the complete bipartite graph K_{n,n} contains a rainbow matching of size n-1. In this talk, we will discuss a multiplicity extension of this conjecture, proposed recently by Anastos, Fabian, Müyesser, and Szabó, and prove a special case of it. More precisely, we will show that, for all positive integers n,a_1, a_2, a_3 satisfying a_1+a_2+a_3 = n-1, every bipartite graph on 2n vertices that is the union of three perfect matchings M_1, M_2, and M_3 contains a matching M such that |M ∩ M_i| =a_i for i ∈ {1,2,3}. This is joint work with Micha Christoph and Tibor Szabó.