Counting restricted orientations of random graphs

Abstract

We count orientations of G(n, p) avoiding certain classes of oriented graphs. In particular, we study T r (n, p), the number of orientations of the binomial random graph G(n, p) in which every copy of K r is transitive, and S r (n, p), the number of orientations of G(n, p) containing no strongly connected copy of K r . We give the correct order of growth of log T r (n, p) and log S r (n, p) up to polylogarithmic factors; for orientations with no cyclic triangle, this significantly improves a result of Allen, Kohayakawa, Mota, and Parente. We also discuss the problem for a single forbidden oriented graph, and state a number of open problems and conjectures. KEYWORDS restricted orientations; random graphs; forbidden digraphs 1 Random Struct Alg. 2020;56:1016-1030.wileyonlinelibrary.com/journal/rsa

Counting orientations of random graphs with no directed k‐cycles

“…They proved that, for every k ⩾ 3, with high probability as n → ∞ we have log 2 D(G(n, p), C ↻ k ) = o(pn 2 ) for p ≫ n −1+1∕(k−1) , and log 2 D(G(n, p), 1) . This result was improved in the case of triangles by Kohayakawa, Morris and the last two authors [6], who proved, among other things, the following result 1 .…”

“…for every k ⩾ 3. A first step towards determining log D(G(n, p), C ↻ k ) for k ⩾ 4 was also given in [6], where it was proved that log D(G(n, p),…”

“…We will count separately the orientations which are 𝓁-locally dense and the orientations which are not r-locally dense but are (r − 1)-locally dense for some 2 ⩽ r ⩽ 𝓁. To count the C ↻ 𝓁+2 -free, 𝓁-locally dense orientations ⃗ G (see Lemma 3.6 (ii)), we proceed similarly to the proof in [6]: . This allows us to prove that the largest independent set in N G (v) has size roughly (log n)∕p 1∕𝓁 , which will be enough to finish the proof of this case using the graph container lemma.…”

Counting H-free orientations of graphs

“…As an aside, we remark that the number of H-free orientations of a random graph G = G(n, p) has also been studied for various choices of H, e.g. for H = C k (see [1,10]). Recently, Araújo, Botler and Mota [4] determined D(n, C 3 ) for all values of n and asked what happens if H is a directed cycle of arbitrary length, even if we are only interested in the case of large n. Our first result is an exact answer to their question for odd cycles.…”

Counting $H$-free orientations of graphs

“…As an aside, we remark that the number of H-free orientations of a random graph G = G(n, p) has also been studied for various choices of H, e.g. for H = C k (see [1,10]). Recently, Araújo, Botler and Mota [4] determined D(n, C 3 ) for all values of n and asked what happens if H is an arbitrary strongly connected directed cycle, even if we are only interested in the case of large n. Our first result is an exact answer to their question for odd cycles.…”