The odd-even invariant and Hamiltonian circuits in tope graphs

Questions of the existence of Hamiltonian circuits in the tope graphs of central arrangements of hyperplanes are considered. Connections between the existence of Hamiltonian circuits in the arrangement and the odd-even invariant of the arrangement are described. Some new results concerning bounds on the odd-even invariant are obtained. All results can be formulated more generally for oriented matroids and are still valid in that setting.