Spanning quadrangulations of triangulated surfaces

  • André Kündgen
  • Carsten Thomassen


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.


Vertex-coloring Edge-coloring Quadrangulation Alternating cycle 

Mathematics Subject Classification

05C10 05C15 


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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

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