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Distant 2-Colored Components on Embeddings Part I: Connecting Faces

Joshua Nevin

Abstract

This is the first 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 $2^{Ω(g)}$ and the precolored components are of distance $2^{Ω(g)}$ 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.

Distant 2-Colored Components on Embeddings Part I: Connecting Faces

Abstract

This is the first 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 and the precolored components are of distance 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.
Paper Structure (12 sections, 27 theorems, 3 equations, 9 figures)

This paper contains 12 sections, 27 theorems, 3 equations, 9 figures.

Table of Contents

  1. Background and Motivation
  2. Conventions of this Paper
  3. The Main Results and Structure of this paper
  4. Rainbows
  5. Link Colorings
  6. Enclosures
  7. The Proof of Theorem \ref{['2207.12531v3.tar:MainCollarResultCutProc']}
  8. Coloring and Deleting Paths
  9. Dealing with paths of length four
  10. Dealing with paths of length two
  11. Completing the proof of Theorem \ref{['2207.12531v3.tar:MainResultColorAndDeletePaths']}
  12. The proof of Theorems \ref{['2207.12531v3.tar:FaceConnectionMainResult']} and \ref{['2207.12531v3.tar:SingleFaceConnRes']}

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

Figures (9)

  • Figure 1.1:
  • Figure 7.1: The ellipse enclosing the diagram represents the cycle $C$
  • Figure 7.2: The ellipse enclosing the diagram represents the cycle $C$
  • Figure 7.3: The ellipse enclosing the diagram represents the cycle $C$
  • Figure 7.4: The ellipse enclosing the diagram represents the cycle $C$
  • ...and 4 more figures

Theorems & Definitions (62)

  • Claim 1
  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Definition 2.1
  • ...and 52 more