A $4$-choosable graph that is not $(8:2)$-choosable

TL;DR

This paper constructs a specific 4-choosable graph that cannot be scaled to be (8:2)-choosable, providing a counterexample to a long-standing question about graph choosability scaling.

Contribution

It presents the first known counterexample showing that (a:b)-choosability does not necessarily scale proportionally for all graphs.

Findings

A 4-choosable graph that is not (8:2)-choosable.

Counterexample to the scaling conjecture in graph choosability.

Answers a question posed in 1980 by Erdős, Rubin, and Taylor.

Abstract

In 1980, Erd\H{o}s, Rubin and Taylor asked whether for all positive integers $a$, $b$, and $m$, every $(a:b)$-choosable graph is also $(am:bm)$-choosable. We provide a negative answer by exhibiting a $4$-choosable graph that is not $(8:2)$-choosable.