Biclique Covers and Partitions

Trevor Pinto

doi:10.37236/3595

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 bicliques - complete bipartite subgraphs - that are needed to cover (resp. partition) the edges of $G$.The 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.

Highlights

1.1 Biclique covers and partitionsA biclique cover of a graph, G, is a collection of bicliques of G, the union of whose edges is E(G)
A related notion is that of a biclique partition: a collection of bicliques of G, whose edges partition the edges of G
We say that a biclique cover is a k-cover if it contains at most k bicliques

Summary

Biclique covers and partitions

A biclique cover of a (simple) graph, G, is a collection of bicliques (complete bipartite subgraphs) of G, the union of whose edges is E(G). One of the key results in the theory of the local biclique cover/ partition numbers, is that of Lupanov [14] in 1956, stating that for all n, there exists a graph G on n vertices with lbc(G). His result is slightly stronger than stated—he showed the same bound holds if lbc(G) is replaced by w(G), the minimum over all covers, κ, of G of the average number of bicliques in κ to which each vertex belongs. This result was later reproved by Chung, Erdos and Spencer [3] in 1983 and Tuza [19] in 1984, among others.

Subcube Intersection Graph Representations

Relationship between τ and ρ

Bounding bp in terms of bc

Relationship between lbc and lbp