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Distant 2-Colored Components on Embeddings Part III: The General Case

Joshua Nevin

Abstract

This is the third in a sequence of three papers in which we prove the following generalization of Thomassen's 5-choosability theorem: Let $G$ be a finite graph embedded on a surface of genus $g$. Then $G$ can be $L$-colored, where $L$ is a list-assignment for $G$ in which every vertex has a 5-list except for a collection of pairwise far-apart components, each precolored with an ordinary 2-coloring, as long as the face-width of $G$ is at least $2^{Ω(g)}$ and the precolored components are of distance at least $2^{Ω(g)}$ apart. This provides an affirmative answer to a generalized version of a conjecture of Thomassen and also generalizes a result from 2017 of Dvořák, Lidický, Mohar, and Postle about distant precolored vertices. In a previous paper, we proved that the above result holds for a restricted class of embeddings which have no separating cycles of length three or four. In this paper, we use this special case to prove that the result holds in the general case.

Distant 2-Colored Components on Embeddings Part III: The General Case

Abstract

This is the third in a sequence of three papers in which we prove the following generalization of Thomassen's 5-choosability theorem: Let be a finite graph embedded on a surface of genus . Then can be -colored, where is a list-assignment for in which every vertex has a 5-list except for a collection of pairwise far-apart components, each precolored with an ordinary 2-coloring, as long as the face-width of is at least and the precolored components are of distance at least apart. This provides an affirmative answer to a generalized version of a conjecture of Thomassen and also generalizes a result from 2017 of Dvořák, Lidický, Mohar, and Postle about distant precolored vertices. In a previous paper, we proved that the above result holds for a restricted class of embeddings which have no separating cycles of length three or four. In this paper, we use this special case to prove that the result holds in the general case.
Paper Structure (10 sections, 27 theorems, 5 equations)

This paper contains 10 sections, 27 theorems, 5 equations.

Table of Contents

  1. Background and Motivation
  2. Conventions and Structure of this Paper
  3. An Annulus Between Short Facial Cycles
  4. Dealing with the case of Nonsplit Depth $\geq 2$
  5. Completing the proof of Theorem \ref{['Main4CycleAnnulusThm']}
  6. An Edge-Maximality Lemma
  7. Charts and Tilings
  8. Properties of Critical Tilings
  9. Cycle Descendants
  10. Completing the proof of Theorem \ref{['ShortReductionTheoremfirst']}

Key Result

Theorem 1.1

Let $G$ be a planar graph with facial cycle $C$. Let $xy\in E(C)$ and $L$ be a list assignment for $V(G)$ such that each vertex of $G\setminus C$ has a list of size at least five and each vertex of $C\setminus\{x,y\}$ has a list of size at least three, where $xy$ is $L$-colorable. Then $G$ is $L$-co

Theorems & Definitions (94)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 84 more