On the existence of tripartite graphs and n-partite graphs

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.