The odd-even invariant and Hamiltonian circuits in tope graphs

TL;DR

This paper investigates the existence of Hamiltonian circuits in tope graphs of hyperplane arrangements, linking it to the odd-even invariant, and provides bounds on this invariant within a generalized oriented matroid framework.

Contribution

It introduces new connections between Hamiltonian circuits and the odd-even invariant, and extends results to oriented matroids.

Findings

Connections established between Hamiltonian circuits and odd-even invariant.

Bounds on the odd-even invariant are provided.

Results applicable to oriented matroids.

Abstract

In this paper we consider the question of the existence of Hamiltonian circuits in the tope graphs of central arrangements of hyperplanes. Some of the results describe connections between the existence of Hamiltonian circuits in the arrangement and the odd-even invariant of the arrangement. In conjunction with this, we present some results concerning bounds on the odd-even invariant. The results given here can be formulated more generally for oriented matroids and are still valid in that setting.