Advertisement

Spanning quadrangulations of triangulated surfaces

  • André Kündgen
  • Carsten Thomassen
Article

Abstract

In this paper we study alternating cycles in graphs embedded in a surface. We observe that 4-vertex-colorability of a triangulation on a surface can be expressed in terms of spanninq quadrangulations, and we establish connections between spanning quadrangulations and cycles in the dual graph which are noncontractible and alternating with respect to a perfect matching. We show that the dual graph of an Eulerian triangulation of an orientable surface other than the sphere has a perfect matching M and an M-alternating noncontractible cycle. As a consequence, every Eulerian triangulation of the torus has a nonbipartite spanning quadrangulation. For an Eulerian triangulation G of the projective plane the situation is different: If the dual graph \(G^*\) is nonbipartite, then \(G^*\) has no noncontractible alternating cycle, and all spanning quadrangulations of G are bipartite. If the dual graph \(G^*\) is bipartite, then it has a noncontractible, M-alternating cycle for some (and hence any) perfect matching, G has a bipartite spanning quadrangulation and also a nonbipartite spanning quadrangulation.

Keywords

Vertex-coloring Edge-coloring Quadrangulation Alternating cycle 

Mathematics Subject Classification

05C10 05C15 

References

  1. 1.
    Bondy, J.A., Murty, U.S.R.: Graph theory with applications. The MacMillan Press , London, England Ltd. (1976)Google Scholar
  2. 2.
    Hutchinson, J., Richter, R.B., Seymour, P.: Coloring Eulerian triangulations. J. Combin. Theor. Ser. B 84, 225–239 (2002)CrossRefzbMATHGoogle Scholar
  3. 3.
    Jensen, T., Toft, B.: Graph Coloring Problems. John Wiley, New York (1995)zbMATHGoogle Scholar
  4. 4.
    Kündgen, A., Ramamurthi, R.: Coloring face-hypergraphs of graphs on surfaces. J. Combin. Theor. Ser. B 85, 307–337 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Mohar, B.: Coloring Eulerian triangulations of the projective plane. Discrete Math. 244, 339–343 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins University Press, Baltimore, USA (2001)Google Scholar
  7. 7.
    Nakamoto, A., Noguchi, K., Ozeki, K.: Spanning bipartite quadrangulations of even triangulations, manuscript (2015)Google Scholar
  8. 8.
    Thomassen, C.: Planarity and duality of finite and infinite graphs. J. Combin. Theor. Ser. B 29, 244–271 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Thomassen, C.: Color-critical graphs on a fixed surface. J. Combin. Theor. Ser. B 70, 67–100 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Department of MathematicsCalifornia State University San MarcosSan MarcosUSA
  2. 2.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkLyngbyDenmark

Personalised recommendations