Bipartite clique minors in graphs of large Hadwiger number

TL;DR

This paper improves bounds relating the Hadwiger number of a graph to the existence of bipartite minors with large Hadwiger number, establishing near-tight conditions and extending results to topological minors.

Contribution

The authors enhance previous bounds on Hadwiger numbers implying bipartite minors, providing tighter asymptotic results and constructions that demonstrate optimality.

Findings

Improved bound: $h(G) ext{ large} ightarrow$ existence of bipartite minors with large Hadwiger number

Established tightness of the new bounds through constructions

Extended results to topological minors

Abstract

The Hadwiger number $h(G)$ is the order of the largest complete minor in $G$. Does sufficient Hadwiger number imply a minor with additional properties? In [2], Geelen et al showed $h(G)\geq (1+o(1))ct\sqrt{\ln t}$ implies $G$ has a bipartite subgraph with Hadwiger number at least $t$, for some explicit $c\sim 1.276\dotsc$. We improve this to $h(G) \geq (1+o(1))t\sqrt{\log_2 t}$, and provide a construction showing this is tight. We also derive improved bounds for the topological minor variant of this problem.