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Counting orientations of random graphs with no directed k-cycles
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Abstract
For every $k \geq 3$, we determine the order of growth, up to polylogarithmic factors, of the number of orientations of the binomial random graph containing no directed cycle of length $k$. This solves a conjecture of Kohayakawa, Morris and the last two authors.
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In 1974, Erdős posed the following problem. Given an oriented graph H , determine or estimate the maximum possible number of H -free orientations of an n -vertex graph. When H is a tournament, the answer was determined precisely for sufficiently large n by Alon and Yuster. In general, when the underlying undirected graph of H contains a cycle, one can obtain accurate bounds by combining an observation of Kozma and Moran with celebrated results on the number of F -free graphs. As the main contribution of the paper, we resolve all remaining cases in an asymptotic sense, thereby giving a rather complete answer to Erdős’s question. Moreover, we determine the answer exactly when H is an odd cycle and n is sufficiently large, answering a question of Araújo, Botler and Mota.
We count orientations of avoiding certain classes of oriented graphs. In particular, we study , the number of orientations of the binomial random graph in which every copy of is transitive, and , the number of orientations of containing no strongly connected copy of . We give the correct order of growth of and 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.
Let Sk(n) be the maximum number of orientations of an n-vertex graph G in which no copy of Kk is strongly connected. For all integers n, k ≥ 4 where n ≥ 5 or k ≥ 5, we prove that Sk(n) = 2tk - 1(n), where tk-1(n) is the number of edges of the n-vertex (k - 1)-partite Turán graph Tk-1(n). Moreover, we prove that Tk-1(n) is the only graph having 2tk-1(n) orientations with no strongly connected copies of Kk.
In this short survey article, we present an elementary, yet quite powerful,
method of enumerating independent sets in graphs. This method was first
employed more than three decades ago by Kleitman and Winston and has
subsequently been used numerous times by many researchers in various contexts.
Our presentation of the method is illustrated with several applications of it
to `real-life' combinatorial problems. In particular, we derive bounds on the
number of independent sets in regular graphs, sum-free subsets of $\{1, \ldots,
n\}$, and $C_4$-free graphs and give a short proof of an analogue of Roth's
theorem on $3$-term arithmetic progressions in sparse random sets of integers
which was originally formulated and proved by Kohayakawa, \L uczak, and R\"odl.
Let (H) over right arrow be an orientation of a graph H. Alon and Yuster proposed the problem of determining or estimating D(n, m, (H) over right arrow), the maximum number of (H) over right arrow -free orientations a graph with n vertices and m edges may have. We consider the maximum number of (H) over right arrow -free orientations of typical graphs G(n, m) with n vertices and m edges. Suppose (H) over right arrow = C-l(((sic))) is the directed cycle of length l >= 3. We show that if m >> n(1+1/(l-1)), then this maximum is 2(o(m)), while m << n(1+1/(l-1)), the it is 2((1-o(1))m).
We improve N. Alon’s [Isr. J. Math. 73, 247–256 (1991; Zbl 0762.05050)] upper bound for the number of independent sets in regular graphs and extend it to almost regular graphs. We also present an upper bound for the number of independent sets of non-typical size for almost regular graphs and an upper bound of the form 2 (n/2)(1-cδ) , where c is a constant, for the number of independent sets in almost regular δ-expanders.
Let T be a fixed tournament on k vertices. Let D(n,T ) denote the maximum number of orientations of an n-vertex graph that have no copy of T. We prove that \(
D{\left( {n,T} \right)} = 2^{{t_{{k - 1^{{{\left( n \right)}}} }} }}
\) for all sufficiently (very) large n, where t
k−1(n) is the maximum possible number of edges of a graphon n vertices with no K
k
, (determined by Turán’s Theorem). The proof is based on a directed version of Szemerédi’s regularity lemma together with some additional ideas and tools from Extremal Graph Theory, and provides an example of a precise result proved by applying this lemma. For the two possible tournaments with three vertices we obtain separate proofs that avoid the use of the regularity lemma and therefore show that in these cases \(
D{\left( {n,T} \right)} = 2^{{{\left\lfloor {n^{2} /4} \right\rfloor }}}
\) already holds for (relatively) small values of n.
Counting graph orientations with no directed triangles
- P Araújo
- F Botler
- G O Mota
P. Araújo, F. Botler, and G. O. Mota, Counting graph orientations with no directed triangles (2020),
available at arXiv:2005.13091v1. ↑1
- D Kleitman
- K Winston
D. J Kleitman and K. J Winston, On the number of graphs without 4-cycles, Discrete Mathematics
41 (1982), no. 2, 167-172. ↑2, 12