Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distant 2-Colored Components on Embeddings Part II: The Short-Inseparable Case

Joshua Nevin

Abstract

This is the second in a sequence of three papers in which we prove the following generalization of Thomassen's 5-choosability theorem: Let $G$ be a 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 this paper we prove that the above result holds for a restricted class of embeddings, i.e. those embeddings which satisfy certain triangulation conditions and do not have separating cycles of length at most four.

Distant 2-Colored Components on Embeddings Part II: The Short-Inseparable Case

Abstract

This is the second in a sequence of three papers in which we prove the following generalization of Thomassen's 5-choosability theorem: Let be a 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 this paper we prove that the above result holds for a restricted class of embeddings, i.e. those embeddings which satisfy certain triangulation conditions and do not have separating cycles of length at most four.
Paper Structure (15 sections, 36 theorems, 23 equations)

This paper contains 15 sections, 36 theorems, 23 equations.

Table of Contents

  1. Background
  2. Two Black Boxes From JNevinThesisManyFacePartIPaperRevised
  3. The Structure and Main Result of this Paper
  4. Mosaics and Their Properties
  5. Webs of Contractible Cycles
  6. Obstructing Cycles
  7. Generalized Chords of Closed Rings in Critical Mosaics
  8. Generalized Chords of Open Rings in Critical Mosaics
  9. Bands of Open Rings in Critical Mosaics
  10. Rainbows and Consistent Paths
  11. Reductions Near the Rings: Open Rings
  12. Boundary Analysis for Closed Rings
  13. Reductions Near the Rings: Closed Rings
  14. The Face-Width of Critical Mosaics
  15. Completing the Proof of Theorem \ref{['AllMosaicsColIntermRes1-4']}

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 $V(G\setminus C)$ has a list of size at least five and each vertex of $V(C)\setminus\{x,y\}$ has a list of size at least three, where $xy$ is $L$-colorable. Then $G$ is

Theorems & Definitions (92)

  • Claim 1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 1.9
  • ...and 82 more