Tight lower bounds on the number of bicliques in false-twin-free graphs

TL;DR

This paper establishes tight lower bounds on the minimum number of bicliques in certain classes of false-twin-free graphs, advancing understanding of their structural properties.

Contribution

It provides the first tight lower bounds on biclique counts in specific false-twin-free graph subclasses, extending theoretical knowledge in graph structure analysis.

Findings

Tight lower bounds for extless{}C4, false-twin extgreater{}-free graphs

Tight lower bounds for extless{}K3, false-twin extgreater{}-free graphs

Discussion on bounds for general graphs

Abstract

A \emph{biclique} is a maximal bipartite complete induced subgraph of $G$. Bicliques have been studied in the last years motivated by the large number of applications. In particular, enumeration of the maximal bicliques has been of interest in data analysis. Associated with this issue, bounds on the maximum number of bicliques were given. In this paper we study bounds on the minimun number of bicliques of a graph. Since adding false-twin vertices to $G$ does not change the number of bicliques, we restrict to false-twin-free graphs. We give a tight lower bound on the minimum number bicliques for a subclass of $\{C_4$,false-twin$\}$-free graphs and for the class of $\{K_3$,false-twin$\}$-free graphs. Finally we discuss the problem for general graphs.