On the Number of Orientations of Random Graphs with No Directed Cycles of a Given Length

Let ~ H be an orientation of a graph H. Alon and Yuster proposed the problem of determining or estimating D(n,m, ~ H), the maximum number of ~ H-free orientations a graph with n vertices and m edges may have. We consider the maximum number of ~ H-free orientations of typical graphs G(n,m) with n vertices and m edges. Suppose ~ H = C ` is the directed cycle of length ` > 3. We show that if m n 1+1/(`−1), then this maximum is 2o(m), while if m n1+1/(`−1), then it is 2(1−o(1))m. ∗An extended abstract of this work appeared in the Proc. of LAGOS ’11, Latin-American Algorithms, Graphs and Optimization Symposium [A note on counting orientations, Electron. Notes in Discrete Math. 37 (2011), 3–8]. The authors gratefully acknowledge the support of NUMEC/USP, Project MaCLinC/USP, and a CAPES/DAAD PROBRAL project (415/ppp-probral/po/D08/11629, Proj. no. 333/09). †Supported by FAPESP (2010/09555-7). ‡Partially supported by FAPESP (2013/03447-6, 2013/07699-0), CNPq (308509/2007-2, 477203/20124) and the NSF (DMS 1102086). §Supported by FAPESP (2009/06294-0, 2012/00036-2, 2013/20733-2) and CNPq (140882/2009-0, 477203/2012-4). ¶Supported by CNPq (131973/2009-6, 140987/2012-6). the electronic journal of combinatorics 21(1) (2014), #P1.52 1