Spanning quadrangulations of triangulated surfaces

André Kündgen, Carsten Thomassen

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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.
Original languageEnglish
JournalUniversitaet Hamburg. Mathematisches Seminar. Abhandlungen
Issue number2
Pages (from-to)357-368
Publication statusPublished - 2017


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