### Abstract

The separation dimension of a graph G is the smallest natural number k for which the vertices of G can be embedded in R^{k} such that any pair of disjoint edges in G can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family F of total orders of the vertices of G such that for any two disjoint edges of G, there exists at least one total order in F in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on n vertices is Θ(log n). In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree d is at most 2^{9 log ∗ d} d. We also demonstrate that the above bound is nearly tight by showing that, for every d, almost all d-regular graphs have separation dimension at least ⌈d/2⌉.

Original language | English (US) |
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Pages (from-to) | 59-64 |

Number of pages | 6 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 29 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2015 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Bounded degree
- Boxicity
- Linegraph
- Separation dimension

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## Cite this

*SIAM Journal on Discrete Mathematics*,

*29*(1), 59-64. https://doi.org/10.1137/140973013