Small counterexamples to the fat minor conjecture

TL;DR

This paper presents simpler counterexamples to the fat minor conjecture, including specific complete and bipartite graphs, by identifying a 'coarse self-similarity' property in related graphs.

Contribution

It introduces a new 'coarse self-similarity' property that enables constructing smaller counterexamples to the fat minor conjecture.

Findings

Constructed smaller counterexamples such as K_t for t ≥ 6

Identified a 'coarse self-similarity' property in certain graphs

Simplified the understanding of graphs disproving the conjecture

Abstract

We narrow the gap between the family of graphs that do and the family of graphs that do not satisfy the fat minor conjecture by obtaining much simpler counterexamples than were previously known, including $K_t, t \geq 6$ and $K_{s,t}, s,t \geq 4$ and $K_{2,2,2}$. This is achieved by establishing a `coarse self-similarity' property of the graphs used by Nguyen, Scott and Seymour to disprove the `coarse Menger conjecture'. This property may be of independent interest.