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Odd list-coloring of graphs of small Euler genus with no short cycles of specific types

Rishi Balaji, Victoria Khazhinsky, Chun-Hung Liu, Kevin Qin

TL;DR

The paper addresses odd list-coloring for graphs embeddable in the torus or Klein bottle under specific cycle-length restrictions, proving that such graphs are odd $5$-choosable when no cycles of length $3$, $4$, or $6$ occur and no two $5$-cycles share an edge. The authors employ a discharging framework on a 2-cell embedding with carefully defined charge transfer rules to exclude minimal counterexamples, establishing the main result and its corollaries. They show that forbidding $3$, $4$, $6$ cycles (and the edge-sharing constraint on $5$-cycles) suffices to guarantee odd $5$-choosability, and they demonstrate the optimality of the color bound via examples at girth $6$. The work thereby extends the theory of odd colorings to topologically constrained graphs and highlights the interplay between cycle structure and list-coloring on surfaces.

Abstract

Odd coloring is a variant of proper coloring and has received wide attention. We study the list-coloring version of this notion in this paper. We prove that if $G$ is a graph embeddable in the torus or the Klein bottle with no cycle of length 3, 4, and 6 such that no 5-cycles share an edge, then for every function $L$ that assigns each vertex of $G$ a set $L(v)$ of size 5, there exists a proper coloring that assigns each vertex $v$ of $G$ an element of $L(v)$ such that for every non-isolated vertex, some color appears an odd number of times on its neighborhood. In particular, every graph embeddable in the torus or the Klein bottle with no cycle of length 3, 4, 6, and 8 is odd 5-choosable. The number of colors in these results are optimal, and there exist graphs embeddable in those surfaces of girth 6 requiring six or seven colors.

Odd list-coloring of graphs of small Euler genus with no short cycles of specific types

TL;DR

The paper addresses odd list-coloring for graphs embeddable in the torus or Klein bottle under specific cycle-length restrictions, proving that such graphs are odd -choosable when no cycles of length , , or occur and no two -cycles share an edge. The authors employ a discharging framework on a 2-cell embedding with carefully defined charge transfer rules to exclude minimal counterexamples, establishing the main result and its corollaries. They show that forbidding , , cycles (and the edge-sharing constraint on -cycles) suffices to guarantee odd -choosability, and they demonstrate the optimality of the color bound via examples at girth . The work thereby extends the theory of odd colorings to topologically constrained graphs and highlights the interplay between cycle structure and list-coloring on surfaces.

Abstract

Odd coloring is a variant of proper coloring and has received wide attention. We study the list-coloring version of this notion in this paper. We prove that if is a graph embeddable in the torus or the Klein bottle with no cycle of length 3, 4, and 6 such that no 5-cycles share an edge, then for every function that assigns each vertex of a set of size 5, there exists a proper coloring that assigns each vertex of an element of such that for every non-isolated vertex, some color appears an odd number of times on its neighborhood. In particular, every graph embeddable in the torus or the Klein bottle with no cycle of length 3, 4, 6, and 8 is odd 5-choosable. The number of colors in these results are optimal, and there exist graphs embeddable in those surfaces of girth 6 requiring six or seven colors.

Paper Structure

This paper contains 4 sections, 30 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Terminology
  3. Structures of a minimal counterexample
  4. Discharging

Key Result

Theorem 1.1

If $G$ is a graph embeddable in the torus or the Klein bottle with no cycle of length 3, 4, and 6 such that no two 5-cycles share an edge, then $G$ is odd 5-choosable.

Figures (1)

  • Figure 1: Diagrams for the rules (R1)-(R8). The numbers in the parentheses indicate the degree of a vertex or the length of a face. Note that two distinct faces possibly share more than one edge, so the actual situation could be more complicated than what the figures represent.

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • ...and 20 more