A Note on Bounded Biclique Coverings of Complete Graphs

TL;DR

This paper establishes a lower bound and asymptotic formula for the minimum number of bounded biclique graphs needed to cover complete graphs, advancing understanding of graph coverings with size constraints.

Contribution

It introduces a lower bound on vertex-weighted biclique coverings and derives an asymptotic formula for coverings with bounded component size.

Findings

Lower bound on vertex-weighted biclique coverings

Asymptotic formula for biclique coverings with bounded size

Improved understanding of complete graph coverings

Abstract

An undirected biclique $K_{a,b}$ is a graph with vertices partitioned into two sets: a set $A$ containing $a$ vertices and a set $B$ containing $b$ vertices such that every vertex in set $A$ is connected to every vertex in set $B$, and such that no two vertices in the same set have an edge between them. A well-known result is that a minimum of $\lceil \log_2{n} \rceil$ bicliques graphs of any size are needed to edge-cover the complete graph on $n$ vertices. We prove a lower bound on minimum vertex-weighted biclique coverings of the complete graph $n$, and use this to prove an asymptotic formula for the minimum number of bicliques $K_{x,x}$ with bounded component size needed to cover the complete graph on $n$ vertices.