On the Biclique cover of the complete graph

TL;DR

This paper investigates the maximum size of complete graphs that can be covered by bicliques of specific types, establishing upper bounds and exploring special cases, with connections to geometric and combinatorial structures.

Contribution

It provides new upper bounds for the size of graphs admitting biclique covers of a given type and links these covers to cross K-intersection families.

Findings

Derived an upper bound for n(k,d).

Improved bounds in special cases.

Connected biclique covers to cross K-intersection families.

Abstract

Let $K$ be a set of $k$ positive integers. A biclique cover of type $K$ of a graph $G$ is a collection of complete bipartite subgraphs of $G$ such that for every edge $e$ of $G$, the number of bicliques need to cover $e$ is a member of $K$. If $K=\{1,2,..., k\}$ then the maximum number of the vertices of a complete graph that admits a biclique cover of type $K$ with $d$ bicliques, $n(k,d)$, is the maximum possible cardinality of a $k$-neighborly family of standard boxes in $\mathbb{R}^d$. In this paper, we obtain an upper bound for $n(k,d)$. Also, we show that the upper bound can be improved in some special cases. Moreover, we show that the existence of the biclique cover of type $K$ of the complete bipartite graph with a perfect matching removed is equivalent to the existence of a cross $K$-intersection family.