Characterization of Bipartite Degree Sequences Without a Prescribed Partition

Determine whether a given single degree sequence admits a bipartite simple-graph realization when the partition of the sequence into the two bipartition sides is not specified, thereby resolving the general Bigraphic Degree Realization problem (BDR).

Background

The paper distinguishes BDR^P, where the two degree subsequences corresponding to the bipartition sides are provided, from BDR, where only one degree sequence is given and a suitable partition must be found. Gale–Ryser completely characterizes BDR^P, but testing all balanced partitions can be difficult because a sequence may have exponentially many such partitions and different partitions may have different realizability status.

The general BDR problem concerns exact realization by a simple bipartite graph, not by the bipartite multigraph relaxations that are the principal subject of the paper. The authors identify this as a longstanding unresolved problem and do not resolve it in the paper.