The Connection between the Number of Realizations for Degree Sequences and Majorization

The graph realization problem is to find for given nonnegative integers a1, . . . , an a simple graph (no loops or multiple edges) such that each vertex vi has degree ai. Given pairs of nonnegative integers (a1, b1), . . . , (an, bn), (i) the bipartite realization problem ask whether there is a bipartite graph (no loops or multiple edges) such that vectors (a1, ..., an) and (b1, ..., bn) correspond to the lists of degrees in the two partite sets, (ii) the digraph realization problem is to find a digraph (no loops or multiple arcs) such that each vertex vi has indegree ai and outdegree bi. The classic literature provides characterizations for the existence of such realizations that are strongly related to the concept of majorization. Aigner and Triesch (1994) extended this approach to a more general result for graphs, leading to an efficient realization algorithm and a short and simple proof for the Erdős-Gallai Theorem. We extend this approach to the bipartite realization problem and the digraph realization problem. Our main result is the connection between majorization and the number of realizations for a degree list in all three problems. We show: if degree list S majorizes S in a certain sense, then S possesses more realizations than S. We prove that constant lists possess the largest number of realizations for fixed n and a fixed number of arcsm when n divides m. So-called minconvex lists for graphs and bipartite graphs or opposed minconvex lists for digraphs maximize the number of realizations when n does not divide m.