The strong fractional choice number can exceed the choice number

Abstract

Erdős, Rubin and Taylor asked whether every $(a,b)$-choosable graph is $(am,bm)$-choosable for all positive integers $a,b,m$. Dvořák, Hu and Sereni disproved this by constructing a $4$-choosable graph that is not $(8,2)$-choosable, and asked whether a $3$-choosable graph can fail to be $(6,2)$-choosable. As a weakening, Li and Zhu asked whether every $k$-choosable graph is $(kb+1,b)$-choosable for all positive integers $k,b$. Equivalently, Xu and Zhu asked whether $\operatorname{ch}_f^s(G)\le\operatorname{ch}(G)$ for every graph $G$. Here $\operatorname{ch}(G)$ is the choice number of a graph $G$, and $\operatorname{ch}_f^s(G)$ is the strong fractional choice number of $G$, which is defined as the infimum of the real numbers $r$ such that $G$ is $(a,b)$-choosable for all positive integers $a,b$ with $a/b\ge r$. In this paper, we prove that, for all integers $t\ge0$ and $b\ge7t+2$, there is a planar graph $G$ with choice number three that is not $(3b+t,b)$-choosable. For $(b,t)=(2,0)$, our construction gives an affirmative answer to the question of Dvořák, Hu and Sereni. This family answers the questions of Li and Zhu and of Xu and Zhu negatively, and shows that the supremum of $\operatorname{ch}_f^s(G)-\operatorname{ch}(G)$ over graphs of choice number three is at least $1/7$. We further prove that, for all integers $k\ge4$, $t\ge0$ and $b\ge3t+2$, there is a graph $G$ with $\operatorname{ch}(G)=k$ that is not $(kb+t,b)$-choosable. For every $k\ge4$, this establishes the stronger lower bound $1/3$ on the supremum of $\operatorname{ch}_f^s(G)-\operatorname{ch}(G)$. The latter construction also produces simpler versions of the counterexamples of Dvořák, Hu and Sereni. It is worth mentioning that the core gadgets in both constructions are very simple. The former is an eight-vertex graph, and the latter is the six-vertex wheel $W_5$.