Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T15:42:53.595Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
DAVID R. WOOD
Affiliation:
School of Mathematics, Monash University, Melbourne, Australia. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Supported by a Leverhulme Trust Research Fellowship.

‡

Supported by the Australian Research Council.

References

Algor, I. and Alon, N.. The star arboricity of graphs. Discrete Math. 75(1–3) (1989), 11–22.CrossRefGoogle Scholar
Alon, N., Basavaraju, M., Chandran, L. S., Mathew, R. and Rajendraprasad, D.. Separation dimension of bounded degree graphs. SIAM J. Discrete Math. 29(1) (2015), 59–64.CrossRefGoogle Scholar
Alon, N., Basavaraju, M., Chandran, L. S., Mathew, R. and Rajendraprasad, D.. Separation dimension and sparsity. J. Graph Theory 89(1) (2018), 14–25.CrossRefGoogle Scholar
Basavaraju, M., Chandran, L. S., Golumbic, M. C., Mathew, R. and Rajendraprasad, D.. Boxicity and separation dimension. In Graph-theoretic concepts in computer science, vol. 8747 of Lecture Notes in Comput. Sci. (Springer, Cham, 2014), 81–92.Google Scholar
Basavaraju, M., Chandran, L. S., Golumbic, M. C., Mathew, R. and Rajendraprasad, D.. Separation dimension of graphs and hypergraphs. Algorithmica 75(1) (2016), 187–204.CrossRefGoogle Scholar
Bharathi, A. P., De, M. and Lahiri, A.. Circular separation dimension of a subclass of planar graphs. Discrete Math. Theor. Comput. Sci. 19(3) (2017), 8.Google Scholar
Chandran, L. S., Mathew, R. and Sivadasan, N.. Boxicity of line graphs. Discrete Math. 311(21) (2011), 2359–2367.CrossRefGoogle Scholar
Erdös, P. and Szekeres, G.. A combinatorial problem in geometry. Compositio Math. 2 (1935), 463–470.Google Scholar
Erdös, P. and Lovász, L.. Problems and results on 3-chromatic hyper-graphs and some related questions. In Infinite and Finite Sets, vol. 10 of Colloq. Math. Soc. János Bolyai (North-Holland, 1975), 609–627.Google Scholar
Füredi, Z. and Kahn, J.. On the dimensions of ordered sets of bounded degree. Order 3(1) (1986), 15–20.CrossRefGoogle Scholar
Hind, H., Molloy, M. and Reed, B.. Colouring a graph frugally. Combinatorica 17(4) (1997), 469–482.CrossRefGoogle Scholar
Ross, J. Kang and Müller, Tobias. Frugal, acyclic and star colourings of graphs. Discrete Appl. Math. 159(16) (2011), 1806–1814.Google Scholar
Loeb, S. J. and West, D. B.. Fractional and circular separation dimension of graphs. European J. Combin. 69 (2018), 19–35.CrossRefGoogle Scholar
Mitzenmacher, M. and Upfal, E.. Probability and Computing (Cambridge University Press, 2005).CrossRefGoogle Scholar
Molloy, M. and Reed, B.. Asymptotically optimal frugal colouring. J. Combin. Theory Ser. B 100(2) (2010), 226–246.CrossRefGoogle Scholar
Scott, A. and Wood, D. R.. Better bounds for poset dimension and boxicity. To appear in Trans. Amer. Math. Soc. (2018).Google Scholar
Ziedan, E., Rajendraprasad, D., Mathew, R., Golumbic, M. C. and Dusart, J.. The induced separation dimension of a graph. Algorithmica 80(10) (2018), 2834–2848.CrossRefGoogle Scholar