Strong fractional choice number of series-parallel graphs

TL;DR

This paper determines the exact strong fractional choice number for series-parallel graphs with certain girth constraints, revealing a precise relationship between girth and fractional choosability.

Contribution

It establishes the exact strong fractional choice number for classes of series-parallel graphs with girth at least k, for specific values of k, advancing understanding of graph choosability.

Findings

Strong fractional choice number of ${\

Q}_k$ is exactly $2+ rac{1}{q}$ for specified girth values.

Provides exact values for the strong fractional choice number in series-parallel graphs with girth constraints.

Abstract

The strong fractional choice number of a graph $G$ is the infimum of those real numbers $r$ such that $G$ is $(\lceil rm \rceil, m)$-choosable for every positive integer $m$. The strong fractional choice number of a family ${\cal G}$ of graphs is the supremum of the strong fractional choice number of graphs in ${\cal G}$. We denote by ${\cal{Q}}_k$ the class of series-parallel graphs with girth at least $k$. This paper proves that for $k=4q-1, 4q,4q+1, 4q+2$, the strong fractional number of ${\cal{Q}}_k$ is exactly $2+ \frac{1}{q}$.