On sign-invariance graphs of uniform oriented matroids (Q750446)

Two points of an oriented matroid form an invariant pair when they have, in all circuits containing them, either the same sign or different signs. Given an oriented matroid M of rank r defined on an n-set E, the invariance graph of M has E as vertex set and the invariant pairs of M as edges. The authors extend results by Shemer [see: \textit{I. Shemer}, Neighbourly Polytopes, Isr. J. Math. 43, 291-314 (1982; Zbl 0598.05010)]. Their emphasis is on Hamiltonian properties of the invariance graph. They prove also the existence of a representable matroid of rank r with n points where no pair of points is invariant, for n, r such that \(n\geq r+3\geq 6\).