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The strong fractional choice number of triangle-free planar graphs

Xiaolan Hu, Rongxing Xu

Abstract

Let $a,b$ be positive integers with $a\ge b$. A graph $G$ is $(a,b)$-choosable if, for every assignment of lists $L(v)$ of size $a$ to the vertices of $G$, there exists a choice of subsets $C(v)\subseteq L(v)$ with $|C(v)|=b$ for each $v$ such that $C(u)\cap C(v)=\emptyset$ whenever $uv\in E(G)$. We show that every triangle-free planar graph is $(15m,4m)$-choosable for any positive integer $m$. As an immediate consequence, the strong fractional choice number of triangle-free planar graphs is at most $15/4$. This appears to be the first non-trivial upper bound on this parameter for this class of graphs. In particular, the case $m=1$ answers affirmatively a question posed by Jiang and Zhu in [J.~Combin.\ Theory Ser.~B, 2019].

The strong fractional choice number of triangle-free planar graphs

Abstract

Let be positive integers with . A graph is -choosable if, for every assignment of lists of size to the vertices of , there exists a choice of subsets with for each such that whenever . We show that every triangle-free planar graph is -choosable for any positive integer . As an immediate consequence, the strong fractional choice number of triangle-free planar graphs is at most . This appears to be the first non-trivial upper bound on this parameter for this class of graphs. In particular, the case answers affirmatively a question posed by Jiang and Zhu in [J.~Combin.\ Theory Ser.~B, 2019].
Paper Structure (13 sections, 13 theorems, 48 equations, 1 figure)

This paper contains 13 sections, 13 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local reduction operations
  4. Nice subgraphs and associated functions
  5. Two key techniques for constructing reduction schemes
  6. Reducible configurations
  7. Discharging
  8. Final charges of the $4$-faces
  9. Final charges of the vertices
  10. The $6^+$-vertices
  11. The vertices of degree $5$
  12. Proof of Theorem \ref{['main-thm']}
  13. Acknowledgments

Key Result

Theorem 3

Every triangle-free planar graph is $(15m,4m)$-choosable for any positive integer $m$. As a corollary, $3+\frac{1}{17} \leq ch^s_f(\mathcal{P}_3) \leq 3 + \frac{3}{4}$.

Figures (1)

  • Figure 1: Illustration of Rule \ref{['R2']}.

Theorems & Definitions (45)

  • Conjecture 1: Zhu2017
  • Theorem 3
  • Definition 4
  • Remark 1
  • Definition 5
  • Definition 6
  • Remark 2
  • Definition 8
  • Definition 9
  • Lemma 10
  • ...and 35 more