Dense minors and bipartite independence numbers

TL;DR

This paper investigates the structure of $m$-joined graphs, establishing bounds on the existence of dense minors and clique minors depending on the relationship between $m$ and $n$, advancing understanding of graph minors in dense graphs.

Contribution

It proves new bounds on the density and size of minors in $m$-joined graphs, extending previous results to larger $m$ relative to $n$.

Findings

For $m \

m$-joined graphs contain dense minors with density $rac{n}{\

m$-joined graphs contain clique minors of size proportional to $rac{n}{\\sqrt{m \\log m}}$ for larger $m$.

Abstract

A graph $G$ is $m$-joined if there is an edge between every two disjoint $m$-sets of vertices. In this paper, we prove that for any $\varepsilon>0$ and sufficiently large $m, n\in \mathbb{N}$ with $m \le n^{1-\varepsilon}$, every $n$-vertex $m$-joined graph $G$ contains a minor with density $\Omega\!\left(\tfrac{n}{\sqrt{m}}\right)$, which is best possible up to a constant factor. When $m \ge n^{1-\varepsilon}$, we further show that $G$ contains a clique minor of order $\Omega\!\left(\tfrac{n}{\sqrt{m\log m}}\right)$.