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We discuss the length {\red $$L_{c,n}$$} of the longest cycle in a sparse random graph {\red $$G_{n,p},$$ $p=c/n$,} $$c$$ constant. We show that for large $$c$$ there exists a function $f(c)$ such that $$L_{c,n}/n\to f(c)$$ a.s. The function $$f(c)=1-\sum_{k=1}^\infty p_k(c)e^{-kc}$$ where $$p_k{\red (c)}$$ is a polynomial in $$c$$. We are only able to explicitly give the values $$p_1,p_2$$, although we could in principle compute any $$p_k$$. We see immediately that the length of the longest path is also asymptotic to $f(c)n$ w.h.p.
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A scaling limit for the length of the longest cycle in a sparse random digraph
We discuss the length $$\vL_{c,n}$$ of the longest directed cycle in the sparse random digraph $$D_{n,p},p=c/n$$, $$c$$ constant. We show that for large $$c$$ there exists a function $$\vf(c)$$ such that $$\vL_{c,n}/n\to \vf(c)$$ a.s. The function $$\vf(c)=1-\sum_{k=1}^\infty p_k(c)e^{-kc}$$ where $$p_k$$ is a polynomial in $$c$$. We are only able to explicitly give the values $$p_1,p_2$$, although we could in principle compute any $$p_k$$.
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- Award ID(s):
- 1952285
- PAR ID:
- 10320472
- Date Published:
- Journal Name:
- Random structures algorithms
- Volume:
- 60
- ISSN:
- 1098-2418
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1002/rsa.21030
