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Towards Characterization of 5-List-Colorability of Toroidal Graphs

Zdeněk Dvořák, Félix Moreno Peñarrubia

TL;DR

This work investigates explicit obstructions to $5$-choosability on toroidal graphs and their relation to $5$-colorability. It develops and implements two central constructs, cycle-canvases and prism-canvases, and introduces a practical criticality-testing framework that combines the Alon–Tarsi method with targeted reductions and greedy heuristics. The main theorem shows that, for graphs on the torus containing a cyclic system of non-contractible triangles with spacing at most $4$, $5$-choosability-critical graphs coincide with Thomassen’s four $6$-critical toroidal graphs, with $K_7$ appearing as a unique exception outside this regime. The results advance the conjecture that a toroidal graph is $5$-choosable if and only if it is $5$-colorable, and they provide substantial computational data (cycle-canvases up to circumference $14$, prism-canvases for spacing up to $4$) to support this direction; the authors also make their code available for reproducibility and further exploration.

Abstract

Through computer-assisted enumeration, we list minimal obstructions for 5-choosability of graphs on the torus with the following additional property: There exists a cyclic system of non-contractible triangles around the torus where the consecutive triangles are at distance at most four. This condition is satisfied by all previously known obstructions, and we verify that there are no additional obstructions with this property. This supports the conjecture that a toroidal graph is 5-choosable if and only if it is 5-colorable.

Towards Characterization of 5-List-Colorability of Toroidal Graphs

TL;DR

This work investigates explicit obstructions to -choosability on toroidal graphs and their relation to -colorability. It develops and implements two central constructs, cycle-canvases and prism-canvases, and introduces a practical criticality-testing framework that combines the Alon–Tarsi method with targeted reductions and greedy heuristics. The main theorem shows that, for graphs on the torus containing a cyclic system of non-contractible triangles with spacing at most , -choosability-critical graphs coincide with Thomassen’s four -critical toroidal graphs, with appearing as a unique exception outside this regime. The results advance the conjecture that a toroidal graph is -choosable if and only if it is -colorable, and they provide substantial computational data (cycle-canvases up to circumference , prism-canvases for spacing up to ) to support this direction; the authors also make their code available for reproducibility and further exploration.

Abstract

Through computer-assisted enumeration, we list minimal obstructions for 5-choosability of graphs on the torus with the following additional property: There exists a cyclic system of non-contractible triangles around the torus where the consecutive triangles are at distance at most four. This condition is satisfied by all previously known obstructions, and we verify that there are no additional obstructions with this property. This supports the conjecture that a toroidal graph is 5-choosable if and only if it is 5-colorable.
Paper Structure (10 sections, 9 theorems, 1 figure, 1 table)

This paper contains 10 sections, 9 theorems, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Summary of Our Results
  3. Critical Cycle-Canvases
  4. Critical Prism-Canvases
  5. Critical Graphs on the Torus
  6. Techniques
  7. Generating Critical Cycle-Canvases
  8. Testing Graph and Canvas Criticality
  9. The Alon-Tarsi Method
  10. Acknowledgments

Key Result

Theorem 1

Suppose that $G$ is a graph drawn on the torus so that $G$ contains a cyclic system of non-contractible triangles of spacing at most four. Then $G$ is critical for 5-choosability if and only if $G$ is one of the four $6$-critical graphs obtained in thomassentorus.

Figures (1)

  • Figure 1: Candidates for critical prism-canvases of spacing 4.

Theorems & Definitions (18)

  • Theorem 1
  • Conjecture 2
  • Conjecture 3
  • Conjecture 4
  • Lemma 5
  • proof
  • Theorem 6
  • Theorem 7
  • Conjecture 8
  • Theorem 9
  • ...and 8 more