More Counterexamples to the Alon-Saks-Seymour and Rank-Coloring Conjectures

Abstract

The chromatic number $\chi(G)$ of a graph $G$ is the minimum number of colors in a proper coloring of the vertices of $G$. The biclique partition number ${\rm bp}(G)$ is the minimum number of complete bipartite subgraphs whose edges partition the edge-set of $G$. The Rank-Coloring Conjecture (formulated by van Nuffelen in 1976) states that $\chi(G)\leq {\rm rank}(A(G))$, where ${\rm rank}(A(G))$ is the rank of the adjacency matrix of $G$. This was disproved in 1989 by Alon and Seymour. In 1991, Alon, Saks, and Seymour conjectured that $\chi(G)\leq {\rm bp}(G)+1$ for any graph $G$. This was recently disproved by Huang and Sudakov. These conjectures are also related to interesting problems in computational complexity. In this paper, we construct new infinite families of counterexamples to both the Alon-Saks-Seymour Conjecture and the Rank-Coloring Conjecture. Our construction is a generalization of similar work by Razborov, and Huang and Sudakov.