Separation dimension and degree

Alex Scott, David R Wood

Research output: Contribution to journalArticleResearchpeer-review


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

Original languageEnglish
Pages (from-to)549-558
Number of pages10
JournalMathematical Proceedings of the Cambridge Philosophical Society
Issue number3
Publication statusPublished - May 2021


  • 2010 Mathematics Subject Classification: 05C62

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