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arXiv · 1909.06823

Topological bounds for graph representations over any field

Meysam Alishahi·Frédéric Meunier
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Abstract

Haviv ({\em European Journal of Combinatorics}, 2019) has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over $\mathbb{R}$. We show that this holds actually for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over $\mathbb{R}$ -- an important graph invariant from coding theory -- and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries. Related complexity results are also discussed.

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Meysam Alishahi, Frédéric Meunier. 2019-09-15. Topological bounds for graph representations over any field. https://arxiv.org/abs/1909.06823

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