Abstract
LetP denote a polyhedral 2-manifold, i.e. a 2-dimensional cell-complex inR d (d≧3) having convex facets, such that set (P) is homeomorphic to a closed 2-dimensional manifold. LetE be any subset of odd valent vertices ofP, andc E its cardinality. Then for the numberc P(E) of facets containing a vertex ofE the inequality 2c P(E)≧cE+1 is proved. This local combinatorial condition shows that several combinatorially possible types of polyhedral 2-manifolds cannot exist.
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Betke, U., Gritzmann, P. A combinatorial condition for the existence of polyhedral 2-manifolds. Israel J. Math. 42, 297–299 (1982). https://doi.org/10.1007/BF02761410
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DOI: https://doi.org/10.1007/BF02761410