Clean triangulations


A polyhedron on a surface is called a clean triangulation if each face is a triangle and each triangle is a face. LetS p (resp.N p ) be the closed orientable (resp. nonorlentable) surface of genusp. If τ(S) is the smallest possible number of triangles in a clean triangulation ofS, the results are: τ(N 1)=20, τ(S 1)=24, limτ(S p )p −1=4, limτ(N p )p −1=2 forp→∞.

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  1. [1]

    G. A. Dirac: Map colour theorems.Canad. J. Math. 4 (1952), 480–490.

    Google Scholar 

  2. [2]

    G. A. Dirac: Short proof of a map colour theorem.Canad. J. Math. 9 (1957), 225–226.

    Google Scholar 

  3. [3]

    J. Edmonds: A combinatorial representation for polyhedral surfaces (abstract).Notices Amer. Math. Soc. 7 (1960), 646.

    Google Scholar 

  4. [4]

    M. Jungerman, andG. Ringel, Minimum triangulations on orientable surfaces.Acta Math. 145 (1980), 121–154.

    Google Scholar 

  5. [5]

    G. Ringel: Das Geschlecht des vollständigen paaren Graphen.Abh. Math. Sem. Univ. Hamburg. 28 (1965), 139–150.

    Google Scholar 

  6. [6]

    G. Ringel: Wie man die geschlossenen nichtorientierbaren Flächen in mőglichst wenig Dreiecke zerlegen kann.Math. Ann. 130 (1955), 317–326.

    Google Scholar 

  7. [7]

    W. T. Tutte: A census of plane triangulations,Canad. J. Math. 14 (1962), 21–28.

    Google Scholar 

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Hartsfield, N., Ringel, G. Clean triangulations. Combinatorica 11, 145–155 (1991).

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AMS subject classification (1980)

  • 05 C 10