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On hyperbolic virtual polytopes and hyperbolic fans

Gaiane Panina (2006)

Open Mathematics

Hyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov’s uniqueness conjecture: Let K ⊂ ℝ3 be a smooth convex body. If for a constant C, at every point of ∂K, we have R 1 ≤ C ≤ R 2 then K is a ball. (R 1 and R 2 stand for the principal curvature radii of ∂K.) This paper gives a new (in comparison with the previous construction by Y.Martinez-Maure and by G.Panina) series of counterexamples to the conjecture. In particular, a hyperbolic...

On light subgraphs in plane graphs of minimum degree five

Stanislav Jendrol', Tomáš Madaras (1996)

Discussiones Mathematicae Graph Theory

A subgraph of a plane graph is light if the sum of the degrees of the vertices of the subgraph in the graph is small. It is well known that a plane graph of minimum degree five contains light edges and light triangles. In this paper we show that every plane graph of minimum degree five contains also light stars $K_{1, 3}$ and $K_{1, 4}$ and a light 4-path P₄. The results obtained for $K_{1, 3}$ and P₄ are best possible.

Displaying 1 – 8 of 8

On hyperbolic virtual polytopes and hyperbolic fans

On light subgraphs in plane graphs of minimum degree five

On longest circuits in certain non-regular planar graphs

On the chiral Archimedean solids.

On the monotonicity of the volume of hyperbolic convex polyhedra.

On the notion of the combinatorial $p$ -parametric property for polyhedra.

On the number of facets of three-dimensional Dirichlet stereohedra. II: Non-cubic groups.

On the volume of unbounded polyhedra in the hyperbolic space.

Currently displaying 1 – 8 of 8