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orthogonality


Combinatorics

The Double Cap Conjecture ★★

Author(s): Kalai

Conjecture   The largest measure of a Lebesgue measurable subset of the unit sphere of $ \mathbb{R}^n $ containing no pair of orthogonal vectors is attained by two open caps of geodesic radius $ \pi/4 $ around the north and south poles.

Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere

Posted by Jon Noel
updated September 15th, 2015
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Graph Theory » Coloring » Vertex coloring

Circular colouring the orthogonality graph ★★

Author(s): DeVos; Ghebleh; Goddyn; Mohar; Naserasr

Let $ {\mathcal O} $ denote the graph with vertex set consisting of all lines through the origin in $ {\mathbb R}^3 $ and two vertices adjacent in $ {\mathcal O} $ if they are perpendicular.

Problem   Is $ \chi_c({\mathcal O}) = 4 $?

Keywords: circular coloring; geometric graph; orthogonality

Posted by mdevos
updated September 28th, 2009
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