Reference : Chromatic numbers for the hyperbolic plane and discrete analogs
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Physical, chemical, mathematical & earth Sciences : Mathematics
Chromatic numbers for the hyperbolic plane and discrete analogs
Parlier, Hugo mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit]
Petit, Camille [> >]
[en] Mathematics - Combinatorics ; Mathematics - Geometric Topology
[en] We study colorings of the hyperbolic plane, analogously to the Hadwiger-Nelson problem for the Euclidean plane. The idea is to color points using the minimum number of colors such that no two points at distance exactly $d$ are of the same color. The problem depends on $d$ and, following a strategy of Kloeckner, we show linear upper bounds on the necessary number of colors. In parallel, we study the same problem on $q$-regular trees and show analogous results. For both settings, we also consider a variant which consists in replacing $d$ with an interval of distances.
23 pages, 5 figures

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