Rainbow matchings in bipartite multigraphs

Suppose that k is a non-negative integer and a bipartite multigraph G is the union of N = ⌊ k + 2 k + 1 n ⌋ − (k + 1) matchings M1, . . . ,MN , each of size n. We show that G has a rainbow matching of size n− k, i.e. a matching of size n− k with all edges coming from different Mi’s. Several choices of parameters relate to known results and conjectures. Suppose that a multigraph G is given with a proper N -edge coloring, i.e. the edge set of G is the union of N matchings M1, . . . , MN . A rainbow matching is a matching whose edges are from different Mi’s. A well-known conjecture of Ryser [8] states that for odd n every 1-factorization of Kn,n has a rainbow matching of size n. The companion conjecture, attributed to Brualdi [3] and Stein [10] states that for every n, every 1-factorization of Kn,n has a rainbow matching of size at least n− 1. These conjectures are known to be true in an asymptotic sense, i.e. every 1-factorization of Kn,n has a rainbow matching containing n − o(n) edges. For the o(n) term, Woolbright [11] and independently Brouwer et al. [4] proved √ n. Shor [9] improved this to 5.518(log n), an error was corrected in [6]. ∗Research was supported in part by OTKA K104373. †Research was supported in part by OTKA K104373.