The Number of Dependent Arcs in an Acyclic Orientation

Let G be a graph with n nodes, e edges, chromatic number and girth g. In an acyclic orientation of G, an arc is dependent if its reversal creates a cycle. It is well known that if < g, then G has an acyclic orientation without dependent arcs. Edelman showed that if G is connected, then every acyclic orientation has at most e ? n + 1 dependent arcs. We show that if G is connected and < g, then G has an acyclic orientation with exactly d dependent arcs for all d e ? n + 1. We also give bounds on the minimum number of dependent arcs in graphs with g. Given an acyclic orientation of a graph, Edelman (as quoted in West 6]) deened an arc to be dependent if its reversal creates a cycle. We consider the number of dependent arcs in an acyclic orientation of a graph. Given a graph G, let n(G) be the number of nodes, let e(G) be the number of edges, and let d max (G) be the maximum number of dependent arcs over all acyclic orientations of G. Edelman proved the following.