Degree Realization by Bipartite Multigraphs

TL;DR

This work extends degree sequence realization theory to bipartite multigraphs by introducing two relaxation measures—total multiplicity and maximum multiplicity—and by analyzing both partitioned (BDR^P) and unpartitioned (BDR) variants. It delivers a Gale–Ryser–style characterization for bipartite multigraph realizations with a fixed partition and bounded total multiplicity, and shows that optimizing the two metrics can yield different realizations. The paper proves NP-hardness for single-sequence bipartite multigraph realizations and provides an output-sensitive algorithm to enumerate all balanced partitions, with polynomial-time results when the number of partitions is small. It also presents sufficient conditions based on the largest degree to guarantee realizability, and compares behavior under general and bipartite graph constraints to highlight the practical implications for network design and related applications.

Abstract

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 sequences that can be realized by a bipartite graph has two variants. In the simpler one, termed BDR$^P$, the partition of the 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 also two variants. For BDR$^P$, where the partition is given as part of the input, a characterization was known for determining whether 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 characterization for determining if there is a bipartite multigraph realization such that the total number of excess edges is at most $t$. We show that optimizing these two measures may lead to different realizations, and that optimizing by one measure may increase the other substantially. 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. Moreover, we show that this hardness result extends to any graph family which is a sub-family of bipartite graphs and a super-family of paths. 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.

Figures (2)

• Figure 1: Optimal multigraph realizations for the sequence $\hat{d} = (8^2,4^3)$ ($n=5$). On the left we have $\hbox{\sf TotMult}(H_1) = 4$ and $\hbox{\sf MaxMult}(H_1) = 5$, while on the right we have $\hbox{\sf TotMult}(H_2) = 7$ and $\hbox{\sf MaxMult}(H_2) = 2$.
• Figure 2: Multigraph bipartite realizations for the sequence $\tilde{d} = (6^2,3^4)$. On the left we have $\hbox{\sf TotMult}^{bi}(H_1) = 3$ and $\hbox{\sf MaxMult}^{bi}(H_1) = 4$; In the center we have $\hbox{\sf TotMult}^{bi}(H_2) = 5$ and $\hbox{\sf MaxMult}^{bi}(H_2) = 2$; On the right we have $\hbox{\sf TotMult}^{bi}(H_3) = 4$ and $\hbox{\sf MaxMult}^{bi}(H_3) = 2$.

Theorems & Definitions (56)

• Theorem 1: Owens and Trent owens1967determining
• Theorem 2: Erdös-Gallai EG60
• Theorem 3: Chungphaisan chungphaisan1974conditions
• Theorem 4: Owens and Trent owens1967determining
• proof
• Corollary 5
• Corollary 7
• proof
• Theorem 8: Gale-Ryser gale1957theoremryser1957combinatorial
• Theorem 9: Berge miller2013reduced