Dense minors and bipartite independence numbers
Xia Wang, Donglei Yang
TL;DR
The paper investigates how the bipartite independence number constraint $\alpha^*(G)<m$ (i.e., $m$-joined graphs) guarantees large minors. It develops an expander-based framework to convert many small connected pieces into a dense minor, yielding a density lower bound $\Omega\left(\frac{n}{\sqrt{m}}\right)$ for $m \le n^{1-\varepsilon}$ and, in a complementary regime, a clique minor of order $\Omega\left(\frac{n}{\sqrt{m\log n}}\right)$. A refined parameter choice provides a sharper density bound $\Omega\left(\frac{n}{\sqrt{m}}\right)$ for the entire regime $m \le n^{1-\varepsilon}$ and demonstrates the existence of clique minors of comparable scale. The results are tight up to constants and extend the understanding of minor structure under bipartite independence constraints through an expander-based construction and a connecting-set technique.
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 $Ω\!\left(\tfrac{n}{\sqrt{m}}\right)$, 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 $Ω\!\left(\tfrac{n}{\sqrt{m\log m}}\right)$.