The structure of random graph orders
The random graph order Pn,p is defined by taking a random graph Gn,p on vertex set [n], treating an edge ij with i ≺ j in [n] as a relation i < j, and taking the transitive closure. A post in a partial order is an element comparable with all others. We investigate the occurrence of posts in random graph orders, showing in particular that Pn,p almost surely has posts if np-1e-π2/3p → ∞, but almost surely does not if this quantity tends to 0. If there are many posts, the partial order decomposes as a linear sum of smaller orders, and we use this decomposition to show that many parameters of a random graph order - for instance, the height, the logarithm of the number of linear extensions, and the number of incomparable pairs - behave as normal random variables. For instance, for the height Hn,p, we prove that, for p in an appropriate range, there are functions αH(p) = e(1 + o(1))p and βH(p) such that (Hn,p - αH(p)(n)/√nβH(p)→dN(0, 1).
SIAM Journal on Discrete Mathematics
Bollobás, B., & Brightwell, G. (1997). The structure of random graph orders. SIAM Journal on Discrete Mathematics, 10 (2), 318-335. https://doi.org/10.1137/S0895480194281215