The Lexicographic Method for the Threshold Cover Problem

Threshold graphs are a class of graphs that have many equivalent definitions and have applications in integer programming and set packing problems. A graph is said to have a threshold cover of size $k$ if its edges can be covered using $k$ threshold graphs. Chv\'atal and Hammer, in 1977, defined the threshold dimension $\mathrm{th}(G)$ of a graph $G$ to be the least integer $k$ such that $G$ has a threshold cover of size $k$ and observed that $\mathrm{th}(G)\geq\chi(G^*)$, where $G^*$ is a suitably constructed auxiliary graph. Raschle and Simon~[Proceedings of the Twenty-seventh Annual ACM Symposium on Theory of Computing, STOC '95, pages 650--661, 1995] proved that $\mathrm{th}(G)=\chi(G^*)$ whenever $G^*$ is bipartite. We show how the lexicographic method of Hell and Huang can be used to obtain a completely new and, we believe, simpler proof for this result. For the case when $G$ is a split graph, our method yields a proof that is much shorter than the ones known in the literature.