Finding dense minors using average degree

TL;DR

This paper investigates the densest possible t-vertex minors in graphs with average degree at least t-1, providing bounds on the number of edges such minors contain and exact results for small t.

Contribution

It establishes new bounds on the density of t-vertex minors in graphs with given average degree, advancing understanding related to Hadwiger's conjecture.

Findings

Existence of t-vertex minors with at least (( frac{1}{ ext{constant}}))inom{t}{2} edges

Bound cannot be improved beyond (( ext{constant}))inom{t}{2} edges

Exact number of edges guaranteed for t  6

Abstract

Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible $t$-vertex minor in graphs of average degree at least $t-1$. We show that if $G$ has average degree at least $t-1$, it contains a minor on $t$ vertices with at least $(\sqrt{2}-1-o(1))\binom{t}{2}$ edges. We show that this cannot be improved beyond $\left(\frac{3}{4}+o(1)\right)\binom{t}{2}$. Finally, for $t\leq 6$ we exactly determine the number of edges we are guaranteed to find in the densest $t$-vertex minor in graphs of average degree at least $t-1$.