Random graphs and covering graphs of posets
For a graph G, we define c(G) to be the minimal number of edges we must delete in order to make G into a covering graph of some poset. We prove that, if p=n-1+η(n),where η(n) is bounded away from 0, then there is a constant k0>0 such that, for a.e. Gp, c(Gp)≥k0n1+η(n).In other words, to make Gp into a covering graph, we must almost surely delete a positive constant proportion of the edges. On the other hand, if p=n-1+η(n), where η(n)→0, then c(Gp)=o(n1+η(n)), almost surely. © 1986 D. Reidel Publishing Company.
Bollobás, B., Brightwell, G., & Nešetřil, J. (1986). Random graphs and covering graphs of posets. Order, 3 (3), 245-255. https://doi.org/10.1007/BF00400288