On the size of induced acyclic subgraphs in random digraphs

Abstract

Let D ∈ D(n,p) denote a simple random digraph obtained by choosing each of the (n choose 2) undirected edges independently with probability 2p and then orienting each chosen edge independently in one of the two directions with equal probability 1/2. Let mas(D) denote the maximum size of an induced acyclic subgraph in D. We obtain tight concentration results on the size of mas(D). Precisely, we show that mas(D) ≤ 2 (ln np + 3e)/ (ln (1-p) -1)almost surely, provided p ≥ W/n for some fixed constant W. This combined with known and new lower bounds shows that (for p satisfying p=ω(1/n) and p ≤0.5) mas(D) = (2 ln np)/ ln (1-p) -1)(1 ±o(1)). This proves a conjecture stated by Subramanian in 2003 for those p such that p = ω(1/n). Our results are also valid for the random digraph obtained by choosing each of the n(n-1) directed edges independently with probability p.