Abstract: A halving of a multigraph G is a decomposition into multigraphs G1 and G2 such that for each vertex v, degree(v,G1) = degree(v,G2). It is easy to see that when G has an euler tour and an even number of edges, we can obtain a halving of G from the tour by alternately putting edges in G1 and G2. In a well-spread halving of a multigraph, we similarly seek to split parallel edges evenly between G1 and G2. Well spread halvings have natural applications to sports scheduling.
In this talk we discuss extending this idea to directed multigraphs, again with an eye toward sports scheduling. Surprisingly, the directed case is in some ways harder, and in some ways easier, than the undirected case.