Acyclic Orientations of Random Graphs

An acyclic orientation of an undirected graph is an orientation of its edges such that the resulting directed graph contains no cycles. The random graphG"n","pis a probability space consisting of subgraphs ofK"nthat are obtained by selecting eachK"n-edge with independent probabilityp. The random graphQ^n"2","pis defined analogously and consists of subgraphs of then-cube,Q^n"2. 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(G"n","p) that counts the number of acyclic orientations ofG"n","p. Finally we study the distribution ofa(G"n","p) anda(Q^n"2","p) and show that log"2a(G"n","p) and log"2a(Q^n"2","p) are sharply concentrated at their respective expectation values.