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$5$-list-coloring toroidal $6$-regular triangulations in linear time

Niranjan Balachandran, Brahadeesh Sankarnarayanan

TL;DR

The paper addresses the problem of 5-list-coloring toroidal $6$-regular triangulations and proves a linear-time algorithm for a broad class of these graphs, while establishing non-$3$-choosability for the same class. It leverages Altshuler's $T(r,s,t)$ parametrization of toroidal triangulations and introduces a list-calculus framework that propagates residual lists across columns to reduce configurations to a finite set, enabling efficient coloring. Theorem 1 identifies specific parameter regimes (e.g., $r\ge 4$, $s\ge 3$; $T(1,s,2)$ with $s\ge9$, $s\neq11$; $T(2,s,t)$ with $s,t$ even) where $5$-list-colorings can be produced in linear time and not $3$-choosability holds, with corollaries including an infinite family that is $5$-chromatic-choosable and linear-time colorable. These results advance algorithmic understanding of list colorings on surfaces, provide practical coloring procedures for a broad class of toroidal triangulations, and lay out precise open questions for the remaining, more exceptional cases.

Abstract

We give an explicit procedure for $5$-list-coloring a large class of toroidal $6$-regular triangulations in linear time. We also show that these graphs are not $3$-choosable.

$5$-list-coloring toroidal $6$-regular triangulations in linear time

TL;DR

The paper addresses the problem of 5-list-coloring toroidal -regular triangulations and proves a linear-time algorithm for a broad class of these graphs, while establishing non--choosability for the same class. It leverages Altshuler's parametrization of toroidal triangulations and introduces a list-calculus framework that propagates residual lists across columns to reduce configurations to a finite set, enabling efficient coloring. Theorem 1 identifies specific parameter regimes (e.g., , ; with , ; with even) where -list-colorings can be produced in linear time and not -choosability holds, with corollaries including an infinite family that is -chromatic-choosable and linear-time colorable. These results advance algorithmic understanding of list colorings on surfaces, provide practical coloring procedures for a broad class of toroidal triangulations, and lay out precise open questions for the remaining, more exceptional cases.

Abstract

We give an explicit procedure for -list-coloring a large class of toroidal -regular triangulations in linear time. We also show that these graphs are not -choosable.

Paper Structure

This paper contains 26 sections, 17 theorems, 11 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. k-choosability is computationally hard
  4. L-coloring is algorithmically hard
  5. Our work
  6. Related work
  7. Colorability vs. choosability
  8. Choosability of grids
  9. Recent algorithmic advances for list colorings of graphs on surfaces
  10. Preliminaries
  11. Outline of the proof of Theorem 1
  12. Preparation for the proof of case (1) in Theorem 1
  13. Proof of case (1) in Theorem 1
  14. Proofs of cases (2) and (3) in Theorem 1
  15. Proof of case (4) in Theorem 1
  16. ...and 11 more sections

Key Result

Theorem 1

Let $G$ be a simple $6$-regular toroidal triangulation. Then, $G$ is $5$-choosable under any of the following conditions: Moreover, the $5$-list colorings can be given in linear time. Furthermore, none of these graphs are $3$-choosable. Hence, $\mathop{\mathrm{\chi_{\ell}}}\nolimits(G) \in \{ 4, 5 \}$ if any of the cases l1 to l4 hold for $G$.

Figures (17)

  • Figure 1: $G = T(5,6,2)$; the edges between the top and bottom rows are not shown in this and all subsequent figures.
  • Figure 2: Illustration of the sizes of the lists on the vertices at each step for $G = C(3,5)$.
  • Figure 3: Illustrations of configurations \ref{['config-a']} through \ref{['config-c']} in criterion \ref{['criterion2.2']}.
  • Figure 4: The configuration of an isolated component in Lemma \ref{['LI:iso-config']}.
  • Figure 5: Illustration of configurations \ref{['config-I']} through \ref{['config-IV']} of Lemma \ref{['L:without-iso']}.
  • ...and 12 more figures

Theorems & Definitions (32)

  • Theorem 1
  • Corollary 2
  • Definition 3
  • Theorem 4: Altshuler Altshuler1973, 1973
  • Lemma 5
  • Definition 6
  • Theorem 7: Cole--Ost--Schirra ColeOstEtAl2001, 2001
  • Definition 8
  • Definition 9
  • Lemma 10: Bondy--Bopanna--Siegel AlonTarsi1992, 1992
  • ...and 22 more