Degree Realization by Bipartite Multigraphs.

The problem of realizing a given degree sequence by a multigraph can be thought of as a relaxation of the classical degree realization problem (where the realizing graph is simple). This paper concerns the case where the realizing multigraph is required to be bipartite. The problem of characterizing degree sequences that can be realized by a bipartite (simple) graph has two variants. In the simpler one, termed BDR, the partition of the degree sequence into two sides is given as part of the input. A complete characterization for realizability in this variant was given by Gale and Ryser over sixty years ago. However, the variant where the partition is not given, termed BDR, is still open. For bipartite multigraph realizations, there are again two variants. For BDR, where the partition is given as part of the input, a complete characterization was known for determining whether the bi-sequence is r -max-bigraphic, namely, if there is a multigraph realization whose underlying graph is bipartite, such that the maximum number of copies of an edge is at most r . We present a complete characterization for determining if there is a bipartite multigraph realization such that the total number of excess edges is at most t . As for the variant BDR, where the partition is not given, we show that determining whether a given (single) sequence admits a bipartite multigraph realization is NP-hard. On the positive side, we provide an algorithm that computes optimal realizations for the case where the number of balanced partitions is polynomial, and present sufficient conditions for the existence of bipartite multigraph realizations that depend only on the largest degree of the sequence.