On the existence of tripartite graphs and n-partite graphs

TL;DR

This paper extends classical degree sequence characterization results from bipartite to tripartite and n-partite graphs, providing necessary and sufficient conditions for their realizability.

Contribution

It generalizes Gale and Ryser's bipartite degree sequence characterization to n-partite graphs, including new necessary and sufficient conditions for tripartite graphs.

Findings

Provides necessary condition for tripartite degree sequences.

Provides sufficient condition for tripartite degree sequences.

Offers a stronger necessary condition for realizability.

Abstract

The degree sequence of a graph is the sequence of the degrees of its vertices. If $\pi$ is a degree sequence of a graph $G$, then $G$ is a realization of $\pi$ and $G$ realizes $\pi$. Determining when a sequence of positive integers is realizable as a degree sequence of a simple graph has received much attention. One of the early results, by Erd\"{o}s and Gallai, characterized degree sequences of graphs. The result was strengthened by Hakimi and Havel. Another generalization is derived by Cai et al. Hoogeveen and Sierksma listed seven criteria and gave a uniform proof. In addition, Gale and Ryser independently established a characterization by using network flows. We extend Gale and Ryser's results from bipartite graphs to tripartite graphs and even $n$-partite graphs. As corollaries, we give a necessary condition and a sufficient condition for the triple $(\sigma_1, \sigma_2, \sigma_3)$ to be realizable by a tripartite graph, where $\sigma_1$, $\sigma_2$ and $\sigma_3$ are all non-increasing sequences of nonnegative integers. We also give a stronger necessary condition for $(\sigma_1, \sigma_2, \sigma_3)$ to be realizable by a tripartite graph.