Coloring even-faced graphs in the torus and the Klein bottle

Abstract

We prove that a triangle-free graph drawn in the torus with all faces bounded by even walks is 3-colorable if and only if it has no subgraph isomorphic to the Cayley graph C(Z 13; 1,5). We also prove that a non-bipartite quadrangulation of the Klein bottle is 3-colorable if and only if it has no non-contractible separating cycle of length at most four and no odd walk homotopic to a non-contractible two-sided simple closed curve. These results settle a conjecture of Thomassen and two conjectures of Archdeacon, Hutchinson, Nakamoto, Negami and Ota.

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Correspondence to Daniel Král’.

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Institute for Theoretical Computer Science is supported as project 1M0545 by the Ministry of Education of the Czech Republic. The author was visiting Georgia Institute of Technology as a Fulbright scholar in the academic year 2005/06.

Partially supported by NSF Grants No. DMS-0200595 and DMS-0354742.

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Král’, D., Thomas, R. Coloring even-faced graphs in the torus and the Klein bottle. Combinatorica 28, 325–341 (2008). https://doi.org/10.1007/s00493-008-2315-z

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Mathematics Subject Classification (2000)

  • 05C15
  • 05C10