Hyperplanes Arrangements and Graph Orientations
by
Daniel Slilaty
Binghamton University

Let G be a graph on vertices x₁, ... , x_n. For each edge (x_i, x_j) in G, the linear equation x_i=x_j defines a hyperplane in Rⁿ. Call the arrangement of these hyperplanes H_G. It is known that the regions defined by H_G are in one-to-one correspndence with acycylic orientations of G. (An orientation of G is an assignment of directional arrows to the edges of G. It is acyclic if no circle (i.e. circuit) in G has cohernt directional arrows.) This provides a nice graph-theoretical description of the regions of an arrangement of hyperplanes. It is also known that this graph-theoretical description is sufficient to represent all orientations of the matroid associated with H_G.

Consider a graph G along with a labeling f of the directed edges of G with nonzero real numbers in which f(x_i, x_j)=1/f(x_j, x_i). For each edge (x_i, x_j) in G, the linear equation x_i=f(x_i, x_j)x_j defines a hyperplane in Rⁿ, call the arrangement of these hyperplanes H_{G, f}. We will give a graph-theoretical description of the regions defined by H_{G, f}. We will also show that this graph-theoretical description, in a number of cases but not in general, suffices to describe all orientations of the matroid associated with H_{G, f}.

Date received: May 31, 1999