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. Chvátal 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.