In: Geometriae Dedicata, 2020, vol. 206, no. 1, p. 151–179
Let Sg,n be a surface of genus g>1 with n>0 punctures equipped with a complete hyperbolic cusp metric. Then it can be uniquely realized as the boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a new variational proof of this result. We also give an alternative proof of the existence and uniqueness of a hyperbolic polyhedral metric with prescribed curvature in a...
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In: Discrete Applied Mathematics, 2020, vol. 276, p. 115–120
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In: Discrete Mathematics, 2018, vol. 341, no. 11, p. 3123–3133
The chromatic number of a subset of Euclidean space is the minimal number of colors sufficient for coloring all points of this subset in such a way that any two points at the distance 1 have different colors. We give new upper bounds on chromatic numbers of spheres. This also allows us to give new upper bounds on chromatic numbers of any bounded subsets.
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In: Mathematical Notes, 2019, vol. 105, no. 5–6, p. 874–880
Borsuk’s celebrated conjecture, which has been disproved, can be stated as follows: in ℝn, there exist no diameter graphs with chromatic number larger than n + 1. In this paper, we prove the existence of counterexamples to Borsuk’s conjecture which, in addition, have large girth. This study is in the spirit of O’Donnell and Kupavskii, who studied the existence of distance graphs with...
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