Acyclic Orientations of Random Graphs

AbstractAn acyclic orientation of an undirected graph is an orientation of its edges such that the resulting directed graph contains no cycles. The random graphGn,pis a probability space consisting of subgraphs ofKnthat are obtained by selecting eachKn-edge with independent probabilityp. The random graphQn2,pis defined analogously and consists of subgraphs of then-cube,Qn2. In this paper we first derive a bijection between certain equivalence classes of permutations and acyclic orientations. Second, we present a lower and an upper bound on the r.v.a(Gn,p) that counts the number of acyclic orientations ofGn,p. Finally we study the distribution ofa(Gn,p) anda(Qn2,p) and show that log2a(Gn,p) and log2a(Qn2,p) are sharply concentrated at their respective expectation values