Constructing dense graphs with sublinear Hadwiger number

TL;DR

This paper presents explicit constructions of dense graphs with sublinear Hadwiger number using blow-ups of smaller graphs, addressing a longstanding open problem and introducing a fractional variant of the clique minor number.

Contribution

It provides a method to explicitly construct dense graphs with sublinear clique minors, advancing understanding of graph minors and Hadwiger numbers.

Findings

Explicit constructions of dense graphs with sublinear Hadwiger number.

Introduction of a fractional variant of the clique minor number.

Connection between graph blow-ups and minor properties.

Abstract

Mader asked to explicitly construct dense graphs for which the size of the largest clique minor is sublinear in the number of vertices. Such graphs exist as a random graph almost surely has this property. This question and variants were popularized by Thomason over several articles. We answer these questions by showing how to explicitly construct such graphs using blow-ups of small graphs with this property. This leads to the study of a fractional variant of the clique minor number, which may be of independent interest.