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Higher-dimensional generalization of Youngs' theorem and circular colorings

Kengo Enami, Takahiro Matsushita

TL;DR

The article extends Youngs' theorem on quadrangulations by introducing a unified framework based on CW complexes with even $2$-skeletons and $\,k$-fundamental groups. It shows that if an odd closed walk in the $1$-skeleton represents a torsion class in $H_1$, then the circular chromatic number of the $1$-skeleton is bounded below by $2+\frac{2}{k-1}$, implying non-$3$-chromaticity in quadrangulated cases. The work develops the $k$-fundamental group, connects it to $k$-coverings and Lovász-type neighborhood complexes, and computes the $k$-fundamental groups of Borsuk graphs to establish the key obstruction. Collectively, these results extend Youngs' phenomenon to higher dimensions and broader embedding settings, with implications for circular colorings and topological graph theory.

Abstract

In 1996, Youngs proved that any quadrangulation of the real projective plane is not 3-chromatic. This result has been extended in various directions over the years, including to other non-orientable closed surfaces, higher-dimensional analogues of quadrangulations and circular colorings. In this paper, we provide a generalization which yields some of these extensions of Youngs' theorem.

Higher-dimensional generalization of Youngs' theorem and circular colorings

TL;DR

The article extends Youngs' theorem on quadrangulations by introducing a unified framework based on CW complexes with even -skeletons and -fundamental groups. It shows that if an odd closed walk in the -skeleton represents a torsion class in , then the circular chromatic number of the -skeleton is bounded below by , implying non--chromaticity in quadrangulated cases. The work develops the -fundamental group, connects it to -coverings and Lovász-type neighborhood complexes, and computes the -fundamental groups of Borsuk graphs to establish the key obstruction. Collectively, these results extend Youngs' phenomenon to higher dimensions and broader embedding settings, with implications for circular colorings and topological graph theory.

Abstract

In 1996, Youngs proved that any quadrangulation of the real projective plane is not 3-chromatic. This result has been extended in various directions over the years, including to other non-orientable closed surfaces, higher-dimensional analogues of quadrangulations and circular colorings. In this paper, we provide a generalization which yields some of these extensions of Youngs' theorem.

Paper Structure

This paper contains 15 sections, 29 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Youngs' theorem
  3. Various generalizations and refinements
  4. Results
  5. Preliminaries
  6. Graphs
  7. Basic facts from algebraic topology
  8. Homology classes of simple closed curves in surfaces
  9. Chromatic numbers
  10. $k$-fundamental groups
  11. $k$-fundamental groups
  12. $k$-covering maps
  13. Proof of the main theorem
  14. Borsuk graphs
  15. Proof of Theorem \ref{['theorem pi_1']}

Key Result

Theorem 1.1

Any quadrangulation of the real projective plane $\mathbb{R} P^2$ is not $3$-chromatic.

Theorems & Definitions (48)

  • Theorem 1.1: Youngs Youngs
  • Theorem 1.2: Archdeacon et al. AHNNO and Mohar and Seymour MS
  • Theorem 1.3: Kaiser et al. Kaiseretal
  • Theorem 1.4: DeVos et al. DGMVZ
  • Theorem A
  • Theorem B
  • Proposition 2.1: see Hatcher
  • Proposition 2.2: See also Hatcher
  • Theorem 2.3: Hatcher
  • Proposition 2.4: see Hatcher
  • ...and 38 more