Journal article
Spanning quadrangulations of triangulated surfaces
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.
Language: | English |
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Publisher: | Springer Berlin Heidelberg |
Year: | 2017 |
Pages: | 357-368 |
ISSN: | 18658784 and 00255858 |
Types: | Journal article |
DOI: | 10.1007/s12188-016-0172-z |
ORCIDs: | Thomassen, Carsten |