Topological bounds for graph representations over any field

TL;DR

This paper extends topological lower bounds on graph parameters, such as chromatic number and minrank, to all fields, providing a unified framework and improving existing bounds with implications in coding theory.

Contribution

It generalizes topological bounds for graph invariants to all fields and introduces independent representation over matroids for broader applicability.

Findings

Topological bounds apply to all fields for graph parameters.

Improved the bound for the minrank parameter over any field.

Introduced the concept of independent representation over matroids.

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.