The strong fractional choice number of $3$-choice critical graphs

TL;DR

This paper determines the exact strong fractional choice number for all 3-choice critical graphs, advancing understanding of fractional list coloring in graph theory.

Contribution

It provides a complete characterization of the strong fractional choice number for all 3-choice critical graphs, a previously unresolved problem.

Findings

Exact strong fractional choice numbers for all 3-choice critical graphs are established.

The results clarify the fractional coloring properties of critical graphs.

The paper advances the theory of fractional list coloring in graph theory.

Abstract

A graph $G$ is called $3$-choice critical if $G$ is not $2$-choosable but any proper subgraph is $2$-choosable. A graph $G$ is strongly fractional $r$-choosable if $G$ is $(a,b)$-choosable for all positive integers $a,b$ for which $a/b \ge r$. The strong fractional choice number of $G$ is $ch_f^s(G) = \inf \{r: G $ is strongly fractional $r$-choosable$\}$. This paper determines the strong fractional choice number of all $3$-choice critical graphs.