Hadwiger's conjecture for graphs with infinite chromatic number

TL;DR

This paper constructs a connected graph with infinite chromatic number that refutes Hadwiger's conjecture in the context of infinite graphs, showing the conjecture does not extend to such cases.

Contribution

It provides a counterexample demonstrating that Hadwiger's conjecture does not hold for graphs with infinite chromatic number.

Findings

Constructed a connected graph with infinite chromatic number

Showed K_omega is not a minor of the graph

Refuted the extension of Hadwiger's conjecture to infinite graphs

Abstract

We construct a connected graph H such that (1) \chi(H) = \omega; (2) K_\omega, the complete graph on \omega points, is not a minor of H. Therefore Hadwiger's conjecture does not hold for graphs with infinite coloring number.