Finding dense minors using average degree

TL;DR

This work studies the densest possible $t$-vertex minor in graphs with average degree at least $t-1$, motivated by Hadwiger-type questions. The authors prove a lower bound showing the existence of a $t$-vertex minor with at least $(\sqrt{2}-1-o(1))\binom{t}{2}$ edges, and they show this constant cannot be improved beyond $(3/4+o(1))$ via explicit extremal constructions. The paper also provides exact densities for small $t$ (2 through 6) and develops a framework combining neighborhood-degree lemmas, random sampling, and minor-to-subgraph reductions (via $k$-trees and cockade-type graphs) to establish these bounds. Together, these results sharpen our understanding of how average degree controls the edge density of minors and contribute to the broader program surrounding Hadwiger's conjecture. The insights have potential implications for extremal minor theory and related graph-structure questions in dense regimes.

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 $t-1$.

Figures (2)

• Figure 1: Example of a $k$-tree in the proof of \ref{['thm:upper']}: $\left\lceil\frac{t+1}{2}\right\rceil$-th power of a path, here illustrated for $t=8$.
• Figure 2: Examples of graphs $S_{k(s(t),t),r,s(t)}$ in the proof of \ref{['thm:upper']}.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4