The extremal function for disconnected minors

TL;DR

This paper establishes an upper bound on the extremal function for graphs excluding a certain disconnected minor, specifically unions of cycles, confirming previous conjectures and introducing a new density-finding theorem.

Contribution

It proves a new upper bound for the extremal function for disconnected minors, confirming conjectures for unions of cycles, and presents a novel theorem on finding disjoint subgraphs with prescribed densities.

Findings

Proved the bound c(H) ≤ (|V(H)| + comp(H))/2 - 1 for unions of cycles.

Confirmed conjectures of Reed, Wood, Harvey regarding extremal functions.

Developed a new theorem for locating disjoint subgraphs with specific densities in dense graphs.

Abstract

For a graph $H$ let $c(H)$ denote the supremum of $|E(G)|/|V(G)|$ taken over all non-null graphs $G$ not containing $H$ as a minor. We show that $$c(H) \leq \frac{|V(H)|+\mathrm{comp}(H)}{2}-1,$$ when $H$ is a union of cycles, verifying conjectures of Reed and Wood, and Harvey and Wood. We derive the above result from a theorem which allows us to find two vertex disjoint subgraphs with prescribed densities in a sufficiently dense graph, which might be of independent interest.