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Some Extensions of Thomassen's Theorem to Longer Paths

Joshua Nevin

Abstract

Let $G$ be a planar embedding with list-assignment $L$ and outer cycle $C$, and let $P$ be a path of length at most four on $C$, where each vertex of $G\setminus C$ has a list of size at least five and each vertex of $C\setminus P$ has a list of size at least three. In this paper, we prove some results about partial $L$-colorings $φ$ of $C$ with the property that any extension of $φ$ to an $L$-coloring of $\textrm{dom}(φ)\cup V(P)$ extends to $L$-color all of $G$. We use these results in a later sequence of papers to prove some results about list-colorings of high-representativity embeddings on surfaces.

Some Extensions of Thomassen's Theorem to Longer Paths

Abstract

Let be a planar embedding with list-assignment and outer cycle , and let be a path of length at most four on , where each vertex of has a list of size at least five and each vertex of has a list of size at least three. In this paper, we prove some results about partial -colorings of with the property that any extension of to an -coloring of extends to -color all of . We use these results in a later sequence of papers to prove some results about list-colorings of high-representativity embeddings on surfaces.
Paper Structure (16 sections, 15 theorems, 3 equations, 11 figures)

This paper contains 16 sections, 15 theorems, 3 equations, 11 figures.

Table of Contents

  1. Introduction and Motivation
  2. Extending Colorings of 2-Paths
  3. Extending Colorings of the Endpoints of 2-Paths
  4. Extending Colorings of 3-Paths
  5. The Proof of \ref{['LabCrownNonEmpt']}
  6. The proof of \ref{['LabCrownNonEmpt2']}
  7. The Proof of \ref{['LabCrownNonEmpt3']}
  8. The Proof of \ref{['LabCrownNonEmpt4']}
  9. The proof of Theorem \ref{['MainHolepunchPaperResulThm']}
  10. Preliminary Restrictions
  11. All chords of $C$ are incident to one of $q_0, q_1$
  12. Dealing with common neighbors of $z, x_i$
  13. Ruling out a common neighbor of $q_0, z, q_1$
  14. Ruling out a common neighbor to $x_0, x_1$ in $G\setminus C$
  15. Completing the Proof of Theorem \ref{['MainHolepunchPaperResulThm']}
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $G$ be a planar embedding with outer cycle $C$, $L$ be a list-assignment for $V(G)$, and $P:=p_0q_0zq_1p_1$ be a subpath of $C$, where $q_0, q_1$ have no common neighbor in $C\setminus P$, and Then there is a partial $L$-coloring $\phi$ of $V(C)\setminus\{q_0, q_1\}$, where $p_0, p_1, z\in\textnormal{dom}(\phi)$, such that each of $q_0, q_1$ has an $L_{\phi}$-list of size at least three, and

Figures (11)

  • Figure 6.1:
  • Figure 6.2:
  • Figure 6.3:
  • Figure 8.1: \ref{['LabCrownNonEmpt4']} of Theorem \ref{['CombinedT1T4ThreePathFactListThm']} is false if we let $|L(p_0)|=|L(p_1)|=2$
  • Figure 9.1: Theorem \ref{['MainHolepunchPaperResulThm']} is false if $q_0, q_1$ are allowed to have a common neighbor in $V(C\setminus\mathring{P})$
  • ...and 6 more figures

Theorems & Definitions (48)

  • Claim 1
  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1: \ref{['MainHolepunchPaperResulThm']}
  • Theorem 1.5
  • Corollary 1.6
  • Definition 1.7
  • Theorem 1.8
  • ...and 38 more