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Displaying similar documents to “Two signed associahedra.”

All Tight Descriptions of 3-Stars in 3-Polytopes with Girth 5

Oleg V. Borodin, Anna O. Ivanova (2017)

Discussiones Mathematicae Graph Theory

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Lebesgue (1940) proved that every 3-polytope P5 of girth 5 has a path of three vertices of degree 3. Madaras (2004) refined this by showing that every P5 has a 3-vertex with two 3-neighbors and the third neighbor of degree at most 4. This description of 3-stars in P5s is tight in the sense that no its parameter can be strengthened due to the dodecahedron combined with the existence of a P5 in which every 3-vertex has a 4-neighbor. We give another tight description of 3-stars in P5s:...

Combinatorial lemmas for polyhedrons I

Adam Idzik, Konstanty Junosza-Szaniawski (2006)

Discussiones Mathematicae Graph Theory

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We formulate general boundary conditions for a labelling of vertices of a triangulation of a polyhedron by vectors to assure the existence of a balanced simplex. The condition is not for each vertex separately, but for a set of vertices of each boundary simplex. This allows us to formulate a theorem, which is more general than the Sperner lemma and theorems of Shapley; Idzik and Junosza-Szaniawski; van der Laan, Talman and Yang. A generalization of the Poincaré-Miranda theorem is also...