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A scaling limit for the length of the longest cycle in a sparse random graph
Michael Anastos,Alan Frieze +1 more
TLDR
It is seen immediately that the length of the longest path is also asymptotic to $f(c)n$ w.h.s.p.Abstract:
We discuss the length of the longest cycle in a sparse random graph $G_{n,p},p=c/n$. $c$ constant. We show that for large $c$ there is a function $f(c)$ such that $L_n(c)/n\to f(c)$ a.s. The function $f(c)=1-\sum_{k=1}^\infty p_k(c)e^{-kc}$ where $p_k$ is a polynomial in $k$. 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.read more
Citations
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Hamilton Cycles in Random Graphs: a bibliography
TL;DR: An annotated bibliography for the study of Hamilton cycles in random graphs and hypergraphs is provided in this paper, where the authors provide an annotated version of their paper.
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Cycle lengths in sparse random graphs
TL;DR: For the Erdős-Renyi model, the authors of as mentioned in this paper showed that the cycle lengths of all cycles that appear in a random regular graph can simultaneously contain the entire range of cycles in the graph.
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Longest paths in random hypergraphs
Oliver Cooley,Frederik Garbe,Eng Keat Hng,Mihyun Kang,Nicolás Sanhueza-Matamala,Julian Zalla +5 more
TL;DR: The `Pathfinder' algorithm is introduced, a depth-first search algorithm which discovers j-tight paths in a k-uniform hypergraph and it is proved that, in the supercritical case, with high probability this algorithm will find a long $j$-tight path.
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Tur\'an-type problems for long cycles in random and pseudo-random graphs.
TL;DR: The results match the classical result of Woodall on the Tur\'an number of long cycles, and can be seen as its random version, showing that the transference principle holds here as well.
Journal ArticleDOI
A scaling limit for the length of the longest cycle in a sparse random digraph
Michael Anastos,Alan Frieze +1 more
TL;DR: The length L→c,n of the longest directed cycle in the sparse random digraph Dn,p,p=c/n, c constant is discussed and it is shown that for large c there exists a function f→(c) such that L→ c,n/n→f→ (c) a.s.
References
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Journal ArticleDOI
The evolution of random graphs
Journal ArticleDOI
A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs
TL;DR: The method determines the asymptotic distribution of the number of short cycles in graphs with a given degree sequence, and gives analogous formulae for hypergraphs.
MonographDOI
Introduction to random graphs
Alan Frieze,Michał Karoński +1 more
TL;DR: All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.
BookDOI
Probabilistic methods for algorithmic discrete mathematics
TL;DR: The Probabilistic Method and Percolation and the Random Cluster Model: Combinatorial and Algorithmic Problems.
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A scaling limit for the length of the longest cycle in a sparse random graph
Michael Anastos,Alan Frieze +1 more