Abstract
Aplane quadrangulation G is a simple plane graph such that each face ofG is quadrilateral. A (*) -orientation D *(G) ofG is an orientation ofG such that the outdegree of each vertex on ∂G is 1 and the outdegrees of other vertices are all 2, where ∂G denotes the outer 4-cycle ofG. In this paper, we shall show that every plane quadrangulationG has at least one (*)-orientation. We also show that any two (*)-orientations ofG can be transformed into one another by a sequence of 4-cycle reversals. Moreover, we apply this fact toorthogonal plane partitions, which are partitions of a square into rectangles by straight segments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Nakamoto, Irreducible quadrangulations of the torus,J. Combin. Theory Ser, B 67 (1996), 183–201.
A. Nakamoto, Irreducible quadrangulations of the Klein bottle,Yokohama Math. J. 43 (1995), 125–139.
A. Nakamoto, K. Ota andT. Tanuma, Cycle reversions in oriented planar triangulations,Yokohama Math. J. 44 (1997), 123–139.
S. Negami andA. Nakamoto, Diagonal transformations of graphs on closed surfaces,Science Rep. of Yokohama Nat. University Sec. I, No. 40 (1993), 71–97.
Author information
Authors and Affiliations
Additional information
A research fellow of the Japan Society for the Promotion of Science.
Rights and permissions
About this article
Cite this article
Nakamoto, A., Watanabe, M. Cycle reversals in oriented plane quadrangulations and orthogonal plane partitions. J Geom 68, 200–208 (2000). https://doi.org/10.1007/BF01221072
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01221072