Separation dimension and degree
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Abstract The separation dimension of a graph G is the minimum positive integer d for which there is an embedding of G into ℝ d , such that every pair of disjoint edges are separated by some axis-parallel hyperplane. We prove a conjecture of Alon et al. [SIAM J. Discrete Math. 2015] by showing that every graph with maximum degree Δ has separation dimension less than 20Δ, which is best possible up to a constant factor. We also prove that graphs with separation dimension 3 have bounded average degree and bounded chromatic number, partially resolving an open problem by Alon et al. [J. Graph Theory 2018].
2002 ◽
Vol 11
(1)
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pp. 103-111
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1996 ◽
Vol 5
(1)
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pp. 15-28
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2022 ◽
pp. 71-82
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2019 ◽
Vol 28
(5)
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pp. 791-810
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1987 ◽
Vol 109
(4)
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pp. 487-490
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