On the role of the equal partition in degree realization by a bipartite graph

Necessary and sufficient conditions for a pair of integer sequences to be the degree sequences of the two sides of a bipartite graph were established more than six decades ago by Gale and Ryser. In contrast, the general question of deciding whether a single sequence is bigraphic, namely, can be realized by a bipartite graph, is still open. We consider even sequences, in which the multiplicity of any integer in the degree sequence is even. One can always partition an even sequence into two identical sequences, resulting in an equal partition. We show that if a given even sequence d is graphic, then there are only two options: either d is bigraphic, or d is 2-bigraphic, namely, can be realized by a bipartite multigraph with maximum multiplicity 2. For an r-graphic sequence we show that it is t-bigraphic for some t≤2r, and we also show that the analysis is tight, namely that t=2r is possible. In addition, we show that given an r-graphic sequence d, there exists an even sequence d^′ which is similar to d in a well-defined sense such that d^′ is even and r-graphic, and therefore t-bigraphic for some t≤2r.