On the binary and Boolean rank of regular matrices

Abstract

A $0,1$ matrix is said to be regular if all of its rows and columns have the same number of ones. We prove that for infinitely many integers k, there exists a square regular $0,1$ matrix with binary rank k, such that the Boolean rank of its complement is $k^{\tilde{\Omega }(\log k)}$. This settles, in a strong form, a question of Pullman (1988) [27] and a conjecture of Hefner et al. (1990) [18]. The result can be viewed as a regular analogue of a recent result of Balodis et al. (2021) [2], motivated by the clique vs. independent set problem in communication complexity and by the (disproved) Alon-Saks-Seymour conjecture in graph theory. As an application of the produced regular matrices, we obtain regular counterexamples to the Alon-Saks-Seymour conjecture and prove that for infinitely many integers k, there exists a regular graph with biclique partition number k and chromatic number $k^{\tilde{\Omega }(\log k)}$.