On some lower bounds on the number of bicliques needed to cover a bipartite graph

TL;DR

This paper compares three lower bounds on the biclique covering number of bipartite graphs, establishing inequalities among them and providing examples that improve existing bounds, thus advancing understanding of graph covering complexities.

Contribution

It introduces a new inequality involving the jk and fool bounds and generalizes previous results, offering sharper bounds and examples in bipartite graph covering theory.

Findings

jk  fool  fool^\u221e  min_Q (rk Q)^2

Only the first inequality is novel

Provides examples where fool  (rk)^{log_4 6}

Abstract

The biclique covering number of a bipartite graph G is the minimum number of complete bipartite subgraphs (bicliques) whose union contains every edge of G. In this little note we compare three lower bounds on the biclique covering number: A bound jk(G) proposed by Jukna & Kulikov (Discrete Math. 2009); the well-known fooling set bound fool(G); the "tensor-power" fooling set bound fool^\infty(G). We show jk \le fool le fool^\infty \le min_Q (rk Q)^2, where the minimum is taken over all matrices with a certain zero/nonzero-pattern. Only the first inequality is really novel, the third one generalizes a result of Dietzfelbinger, Hromkovi\v{c}, Schnitger (1994). We also give examples for which fool \ge (rk)^{log_4 6} improving on Dietzfelbinger et al.