Publication: Research - peer-review › Journal article – Annual report year: 2017
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.
|Journal||Universitaet Hamburg. Mathematisches Seminar. Abhandlungen|
|Number of pages||12|
|State||E-pub ahead of print - 2017|
|Citations||Web of Science® Times Cited: No match on DOI|