Separation dimension and degree

The separation dimension of a graph G is the minimum positive integer d for which there is an embedding of G into ℝd, 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...

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Published inMathematical proceedings of the Cambridge Philosophical Society Vol. 170; no. 3; pp. 549 - 558
Main Authors SCOTT, ALEX, WOOD, DAVID R.
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.05.2021
Subjects
Graph theory
Hyperplanes
Separation
05C62
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Summary:The separation dimension of a graph G is the minimum positive integer d for which there is an embedding of G into ℝd, 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].
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ISSN:0305-0041
1469-8064
DOI:10.1017/S0305004119000525