Designs and topology

Grannell, Mike and Griggs, Terry (2007). Designs and topology. In: Hilton, A. and Talbot, J. eds. Surveys in Combinatorics 2007. London Mathematical Society Lecture Note Series 346, London Mat (346). Cambridge, UK: Cambridge University Press, pp. 121–174.


An embedding of a graph in a surface gives rise to a combinatorial design whose blocks correspond to the faces of the embedding. Particularly interesting graphs include complete and complete multipartite graphs. Embeddings of these in which the faces are triangles, Hamiltonian cycles, or Eulerian cycles generate interesting designs. These designs include twofold, Mendelsohn and Steiner triple systems, and Latin squares. We examine some of these cases, looking at construction methods, structural properties and enumeration problems.

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