Rotation system

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In combinatorial mathematics, rotation systems encode the local circular ordering of the edges around the vertices, according to a 2-cell embedding of a connected multigraph on a closed orientable surface.

A rotation system (or map, or combinatorial map) is a pair (σ,θ) formed by a permutation σ and a fixed-point free involution θ on a ground set B of darts (or flags, or half-edges) so that the group \langle\sigma,\theta\rangle generated by σ and θ acts transitively on B. The underlying multigraph of the rotation system (σ,θ) has the orbits of σ as its vertices and the orbits of θ as its edges, an edge being incident to a vertex if the corresponding orbits of σ and θ intersect.

Every rotation scheme defines a unique 2-cell embedding of a connected multigraph on a closed orientable surface (up to orientation preserving topological equivalence). Conversely, any embedding of a connected multigraph G on an orientable closed surface defines a unique rotation system having G as its underlying multigraph.

This fundamental theorem has been first settled on a dual form by Heffter and extensively used by Ringel during the 1950s. Independently, Edmonds gave the primal form of the theorem et the details of his study have been popularized by Youngs. The generalization to the whole set of multigraphs has been developed by Gross and Alpert.

The vertices, the edges and the faces of the embedding correspond to the orbits of σ, θ and σθ, respectively. According to Euler formula we deduce the genus g of the closed orientable surface defined by the rotation system (σ,θ) (surface on which the underlying multigraph is 2-cell embedded):

g=1-\frac{1}{2}(|Z(\sigma)|-|Z(\theta)|+|Z(\sigma\theta)|)

where Z(φ) denotes the set of the orbits of permutation φ.

[edit] References

  • R. Cori and A. Machì, Maps, hypermaps and their automorphisms, Expo. Math. 10 (1992), 403–467.
  • J. Edmonds, A combinatorial representation for polyhedral surfaces, Notices of American Mathematical Society 7 (1960), 643.
  • J.L. Gross and S.R. Alpert, The topological theory of current graphs, J. Comb. Theory Ser. B 17 (1974), 218–233.
  • L. Heffter, Über das Problem der Nachbargebiete, Math. Ann. 8 (1891), 17–20.
  • B. Mohar and C. Thomassen, Graphs on Surfaces, The Johns Hopkins University Press (2001).
  • G. Ringel, Das Geschlecht des vollstandingen paaren Graphen, Abh. Math. Sem. Univ. Hamburg 28 (1965), 139–150.
  • J.W.T. Youngs, Minimal imbeddings and the genus of a graph, J. Math. and Mech. 12 (1963), 303–315.
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