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Separation dimension and degree

Part of: Graph theory

Published online by Cambridge University Press:  22 November 2019

ALEX SCOTT  and
DAVID R. WOOD
ALEX SCOTT
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG. e-mail: scott@maths.ox.ac.uk
DAVID R. WOOD
Affiliation:
School of Mathematics, Monash University, Melbourne, Australia. e-mail: david.wood@monash.edu
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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 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].

MSC classification

Primary: 05C62: Graph representations (geometric and intersection representations, etc.)
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 3 , May 2021 , pp. 549 - 558
Copyright
© Cambridge Philosophical Society 2019

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Footnotes

†

Supported by a Leverhulme Trust Research Fellowship.

‡

Supported by the Australian Research Council.

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