Abstract
Let $\vec H$ be an orientation of a graph $H$. Alon and Yuster proposed the problem of determining or estimating $D(n,m,\vec H)$, the maximum number of $\vec H$-free orientations a graph with $n$ vertices and $m$ edges may have. We consider the maximum number of $\vec H$-free orientations of typical graphs $G(n,m)$ with $n$ vertices and $m$ edges. Suppose $\vec H =C^\circlearrowright_\ell $ is the directed cycle of length $\ell\geq 3$. We show that if ${m\gg n^{1+1/(\ell-1)}}$, then this maximum is $2^{o(m)}$, while if ${m\ll n^{1+1/(\ell-1)}}$, then it is $2^{(1-o(1))m}$.
Highlights
Given a simple graph H, an orientation H of H is obtained by assigning an ‘orientation’ or ‘direction’ to each of its edges
We restricted our attention to counting C -free orientations
Using versions of Theorem 2.2 that work for general graphs H, one may prove certain results on the number of H-free orientations of G(n, p) for orientations H of any given graph H
Summary
Given a simple graph H, an orientation H of H is obtained by assigning an ‘orientation’ or ‘direction’ to each of its edges. Such an H is called oriented graph. We investigate a similar problem, where we consider ‘typical’ graphs G(n, m) with n vertices and m edges. Let us state our main theorem in terms of the binomial random graph G(n, p) (one may derive corresponding results for the G(n, m) model from the G(n, p) results below by standard means; the reader is referred to [8, Section 1.4] for this ‘equivalence’ of the models G(n, p) and G(n, m)). Since our results are mostly asymptotic, when convenient, we freely and tacitly suppose that n is larger than a suitably large constant
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
