On the size of induced acyclic subgraphs in random digraphs
Joel Spencer, C. R. Subramanian
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.Full Text: PDF PostScript