Average degree conditions forcing a minor

TL;DR

This paper improves bounds on the average degree needed to force a graph as a minor, especially for sparse graphs with high degree vertices, advancing understanding in graph minor theory.

Contribution

It strengthens a recent result by Reed and Wood, providing better bounds for sparse graphs with many high degree vertices, and generalizes known results for unbalanced complete bipartite graphs.

Findings

Improved bounds on average degree for forcing H-minors

Solved an open problem by Reed and Wood

Generalized results for unbalanced bipartite graphs

Abstract

Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger's Conjecture, much research has focused on the average degree required to force a complete graph as a minor. Subsequently, various authors have consider the average degree required to force an arbitrary graph $H$ as a minor. Here, we strengthen (under certain conditions) a recent result by Reed and Wood, giving better bounds on the average degree required to force an $H$-minor when $H$ is a sparse graph with many high degree vertices. This solves an open problem of Reed and Wood, and also generalises (to within a constant factor) known results when $H$ is an unbalanced complete bipartite graph.