Chromatic numbers of quadrangulations on closed surfaces

Dan Archdeacon, Joan Hutchinson, Atsuhiro Nakamoto, Seiya Negam, Katsuhiro Ota

Research output: Contribution to journalArticle

29 Citations (Scopus)

Abstract

It has been shown that every quadrangulation on any non-spherical orientable closed surface with a sufficiently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface Nk has chromatic number at least 4 if G has a cycle of odd length which cuts open Nk into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface Nk admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity.

Original languageEnglish
Pages (from-to)100-114
Number of pages15
JournalJournal of Graph Theory
Volume37
Issue number2
DOIs
Publication statusPublished - 2001 Jun

Keywords

  • Chromatic number
  • Eulerian quadrangulations
  • Representativity

ASJC Scopus subject areas

  • Geometry and Topology

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    Archdeacon, D., Hutchinson, J., Nakamoto, A., Negam, S., & Ota, K. (2001). Chromatic numbers of quadrangulations on closed surfaces. Journal of Graph Theory, 37(2), 100-114. https://doi.org/10.1002/jgt.1005