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List colouring triangle free planar graphs

Jianzhang Hu, Xuding Zhu

TL;DR

If G is a triangle free planar graph and L is a list assignment of G such that $\mid L(v)mid = 4$ for each vertex of G, then G is L-colourable.

Abstract

This paper proves the following result: Assume $G$ is a triangle free planar graph, $X$ is an independent set of $G$. If $L$ is a list assignment of $G$ such that $\mid L(v)\mid = 4$ for each vertex $v \in V(G)-X$ and $\mid L(v)\mid = 3$ for each vertex $v \in X$, then $G$ is $L$-colourable.

List colouring triangle free planar graphs

TL;DR

If G is a triangle free planar graph and L is a list assignment of G such that for each vertex of G, then G is L-colourable.

Abstract

This paper proves the following result: Assume is a triangle free planar graph, is an independent set of . If is a list assignment of such that for each vertex and for each vertex , then is -colourable.

Paper Structure

This paper contains 4 sections, 18 theorems, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. Some properties of a minimum counterexample
  3. Proof of Theorem
  4. Proof of Lemma \ref{['clm-no4cycle']}

Key Result

Theorem 2

Assume $G$ is a triangle free planar graph and $X$ is an independent set of $G$. If $L$ is a list assignment of $G$ such that $|L(v)|=3$ for $v \in X$ and $|L(v)| = 4$ for $v \in V(G)-X$, then $G$ is $L$-colourable.

Figures (3)

  • Figure : (a)
  • Figure : (a)
  • Figure : (b)

Theorems & Definitions (27)

  • Conjecture 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 3
  • ...and 17 more