Small Minors in Dense Graphs

TL;DR

This paper proves that dense graphs contain small, logarithmically-sized complete graph minors, with precise bounds for small cases and extensions to surface-embedded graphs, advancing understanding of graph minors in dense structures.

Contribution

It establishes bounds on the size of complete graph minors in dense graphs, including logarithmic size bounds and exact thresholds for small minors, extending to surface-embedded graphs.

Findings

Graphs with high average degree contain small $K_t$-models.

Size of $K_t$-models is at most logarithmic in the number of vertices.

Exact bounds for $f(t)$ for small $t$, especially $t extless=4$.

Abstract

A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of the whole graph. More precisely, we describe functions $f$ and $h$ such that every graph with $n$ vertices and average degree at least $f(t)$ contains a $K_t$-model with at most $h(t)\cdot\log n$ vertices. The logarithmic dependence on $n$ is best possible (for fixed $t$). In general, we prove that $f(t)\leq 2^{t-1}+\eps$. For $t\leq 4$, we determine the least value of $f(t)$; in particular $f(3)=2+\eps$ and $f(4)=4+\eps$. For $t\leq4$, we establish similar results for graphs embedded on surfaces, where the size of the $K_t$-model is bounded (for fixed $t$).