Spanning quadrangulations of triangulated surfaces

作者：Kundgen Andre^*; Thomassen Carsten

来源：Abhandlungen aus Dem Mathematischen Seminar der Universitat Hamburg, 2017, 87(2): 357-368.

DOI：10.1007/s12188-016-0172-z

• Vertex-coloring
• Edge-coloring
• Quadrangulation
• Alternating cycle

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 is nonbipartite, then has no noncontractible alternating cycle, and all spanning quadrangulations of G* are bipartite. If the dual graph 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.

• 出版日期2017-10