The Number of Dependent Arcs in an Acyclic Orientation

LetGbe a graph withnnodes,eedges, chromatic numberχ, and girthg. In an acyclic orientation ofG, an arc isdependentif its reversal creates a cycle. It is well known that ifχ<g, thenGhas an acyclic orientation without dependent arcs. Edelman showed that ifGis connected, then every acyclic orientation has at moste−n+1 dependent arcs. We show that ifGis connected andχ<g, thenGhas an acyclic orientation with exactlyddependent arcs for alld⩽e−n+1. We also give bounds on the minimum number of dependent arcs in graphs withχ⩾g.