Systems of distinct representatives

Let F = (S1, …, Sn) be a family of subsets of a finite set S. A sequence F = (f1, …, fn) of elements of S is called a system of representatives of F if fi ∈ Si, for i = 1,2, …, n. If the elements of F are distinct then F is called a system of distinct representatives (SDR) for F.

Example 11.1.1Let S1={u2, u3, u4}, S2={u1, u2, u3} and S3={u3, u4, u5}. Then F = (u2, u1, u3) is an SDR for F = (S1, S2, S3), since u2 ∈ S1, u1 ∈ S2 and u3 ∈ S3.

Many criteria for the existence of systems of representatives, under various restrictions, have been developed (see Mirsky (1971)). Bipartite graphs have proven to be a particularly useful tool in these investigations, since every collection of subsets F can be represented by the bipartite graph G(F) with bipartition (V1, V2) where V1 = {S1, …, Sn}, V2 = S and the vertices Si ∈ V1 and u ∈ V2 are joined by an edge if and only if u ∈ Si. We shall give a few examples of results on SDRs to demonstrate how this representation can be used, which employ a variety of different graph theoretic results. We begin with the principal result in this area, obtained by P. Hall (1935).