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Register a new userCounting restricted orientations of random graphs
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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.
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Let $S_k(n)$ be the maximum number of orientations of an $n$-vertex graph $G$ in which no copy of $K_k$ is strongly connected. For all integers $n$, $kgeq 4$ where $ngeq 5$ or $kgeq 5$, we prove that $S_k(n) = 2^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the number of edges of the $n$-vertex $(k-1)$-partite Turan graph $T_{k-1}(n)$, and that $T_{k-1}(n)$ is the only $n$-vertex graph with this number of orientations. Furthermore, $S_4(4) = 40$ and this maximality is achieved only by $K_4$.
Formulas are derived for counting walks in the Kronecker product of graphs, and the associated spectral distributions are obtained by the Mellin convolution of probability distributions. Two-dimensional restricted lattices admitting the Kronecker product structure are listed, and their spectral distributions are calculated in terms of elliptic integrals.
Alon and Yuster proved that the number of orientations of any $n$-vertex graph in which every $K_3$ is transitively oriented is at most $2^{lfloor n^2/4rfloor}$ for $n geq 10^4$ and conjectured that the precise lower bound on $n$ should be $n geq 8$. We confirm their conjecture and, additionally, characterize the extremal families by showing that the balanced complete bipartite graph with $n$ vertices is the only $n$-vertex graph for which there are exactly $2^{lfloor n^2/4rfloor}$ such orientations.
A graph G is 1-extendable if every edge belongs to at least one 1-factor. Let G be a graph with a 1-factor F. Then an even F-orientation of G is an orientation in which each F-alternating cycle has exactly an even number of edges directed in the same fixed direction around the cycle. In this paper, we examine the structure of 1-extendible graphs G which have no even F-orientation where F is a fixed 1-factor of G. In the case of cubic graphs we give a characterization. In a companion paper [M. Abreu, D. Labbate and J. Sheehan. Even orientations of graphs: Part II], we complete this characterization in the case of regular graphs, graphs of connectivity at least four and k--regular graphs for $kge3$. Moreover, we will point out a relationship between our results on even orientations and Pfaffian graphs developed in [M. Abreu, D. Labbate and J. Sheehan. Even orientations and Pfaffian graphs].
We establish mild conditions under which a possibly irregular, sparse graph $G$ has many strong orientations. Given a graph $G$ on $n$ vertices, orient each edge in either direction with probability $1/2$ independently. We show that if $G$ satisfies a minimum degree condition of $(1+c_1)log_2{n}$ and has Cheeger constant at least $c_2frac{log_2log_2{n}}{log_2{n}}$, then the resulting randomly oriented directed graph is strongly connected with high probability. This Cheeger constant bound can be replaced by an analogous spectral condition via the Cheeger inequality. Additionally, we provide an explicit construction to show our minimum degree condition is tight while the Cheeger constant bound is tight up to a $log_2log_2{n}$ factor.
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