Biclique Covers and Partitions

TL;DR

This paper investigates bounds on biclique cover and partition numbers of graphs, explores local variants, and connects these concepts to subcube intersection graph representations, revealing limitations and open questions.

Contribution

It establishes bounds relating biclique cover and partition numbers, analyzes local variants, and links these concepts to subcube intersection graph representations, highlighting open problems.

Findings

Bound $ ext{bp}(G)$ in terms of $ ext{bc}(G)$ as $rac{1}{2}(3^{ ext{bc}(G)}-1)$.

Show that $ ext{lbp}(G)$ can be arbitrarily large even when $ ext{lbc}(G)=2$.

Reduced the problem of representing graphs as subcube intersection graphs to a known dimension problem.

Abstract

The biclique cover number (resp. biclique partition number) of a graph $G$, $\mathrm{bc}(G$) (resp. $\mathrm{bp}(G)$), is the least number of biclique (complete bipartite) subgraphs that are needed to cover (resp. partition) the edges of $G$. The \emph{local biclique cover number} (resp. local biclique partition number) of a graph $G$, $\mathrm{lbc}(G$) (resp. $\mathrm{lbp}(G)$), is the least $r$ such that there is a cover (resp. partition) of the edges of $G$ by bicliques with no vertex in more than $r$ of these bicliques. We show that $\mathrm{bp}(G)$ may be bounded in terms of $\mathrm{bc}(G)$, in particular, $\mathrm{bp}(G)\leq \frac{1}{2}(3^\mathrm{bc(G)}-1)$. However, the analogous result does not hold for the local measures. Indeed, in our main result, we show that $\mathrm{lbp}(G)$ can be arbitrarily large, even for graphs with $\mathrm{lbc}(G)=2$. For such graphs, $G$, we try to bound $\mathrm{lbp}(G)$ in terms of additional information about biclique covers of $G$. We both answer and leave open questions related to this. There is a well known link between biclique covers and subcube intersection graphs. We consider the problem of finding the least $r(n)$ for which every graph on $n$ vertices can be represented as a subcube intersection graph in which every subcube has dimension $r$. We reduce this problem to the much studied question of finding the least $d(n)$ such that every graph on $n$ vertices is the intersection graph of subcubes of a $d$-dimensional cube.