A Computation of the Expected Number of Posts in a Finite Random Graph Order

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dc.creatorBombelli, Luca
dc.creatorSeggev, Itai
dc.creatorWatson, Sam
dc.date2008-09-12
dc.date2008-09-25
dc.date.accessioned2026-07-07T10:04:51Z
dc.date.available2026-07-07T10:04:51Z
dc.descriptionA random graph order is a partial order achieved by independently sprinkling relations on a vertex set (each with probability $p$) and adding relations to satisfy the requirement of transitivity. A \textit{post} is an element in a partially ordered set which is related to every other element. Alon et al.\ \cite{Alon} proved a result for the average number of posts among the elements $\{1,2,...,n\}$ in a random graph order on $\mathbb{Z}$. We refine this result by providing an expression for the average number of posts in a random graph order on $\{1,2,...,n\}$, thereby quantifying the edge effects associated with the elements $\mathbb{Z}\backslash\{1,2,...,n\}$. Specifically, we prove that the expected number of posts in a random graph order of size $n$ is asymptotically linear in $n$ with a positive $y$-intercept. The error associated with this approximation decreases monotonically and rapidly in $n$, permitting accurate computation of the expected number of posts for any $n$ and $p$. We also prove, as a lemma, a bound on the difference between the Euler function and its partial products that may be of interest in its own right.
dc.description11 pages, 6 figures; version 2 adds missing .bbl file for bibliography
dc.identifierhttps://arxiv.org/abs/0809.2258
dc.identifierhttp://arxiv.org/abs/0809.2258
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/169829
dc.subjectCombinatorics
dc.subjectGeneral Relativity and Quantum Cosmology
dc.subjectMathematical Physics
dc.subject05C80
dc.titleA Computation of the Expected Number of Posts in a Finite Random Graph Order
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