Biclique Covers and Partitions
Trevor Pinto

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 , ) (resp. ), is the least number of biclique (complete bipartite) subgraphs that are needed to cover (resp. partition) the edges of . The \emph{local biclique cover number} (resp. local biclique partition number) of a graph , ) (resp. ), is the least such that there is a cover (resp. partition) of the edges of by bicliques with no vertex in more than of these bicliques. We show that may be bounded in terms of , in particular, . However, the analogous result does not hold for the local measures. Indeed, in our main result, we show that can be arbitrarily large, even for graphs with . For such graphs, , we try…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Graph theory and applications · Advanced Graph Theory Research
