• Corpus ID: 232417743

Paths, cycles and sprinkling in random hypergraphs

@inproceedings{Cooley2021PathsCA,
  title={Paths, cycles and sprinkling in random hypergraphs},
  author={Oliver Cooley},
  year={2021}
}
We prove a lower bound on the length of the longest j-tight cycle in a k-uniform binomial random hypergraph for any 2 ≤ j ≤ k − 1. We first prove the existence of a j-tight path of the required length. The standard “sprinkling” argument is not enough to show that this path can be closed to a j-tight cycle – we therefore show that the path has many extensions, which is sufficient to allow the sprinkling to close the cycle. 

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References

SHOWING 1-9 OF 9 REFERENCES
Note on induced paths in sparse random graphs
We show that for d ≥ d0(ǫ), with high probability, the random graph G(n, d/n) contains an induced path of length (3/2 − ǫ) d log d. This improves a result obtained independently by Łuczak and Suen in
An Improved Upper Bound on the Length of the Longest Cycle of a Supercritical Random Graph
We improve Luczak's upper bounds on the length of the longest cycle in the random graph G(n,M) in the "supercritical phase" where M=n/2+s and s=o(n) but n^{2/3}=o(s). The new upper bound is
The size of the giant high-order component in random hypergraphs
The phase transition in the size of the giant component in random graphs is one of the most well‐studied phenomena in random graph theory. For hypergraphs, there are many possible generalizations of
Largest Components in Random Hypergraphs
TLDR
It is shown that the existence of a j-tuple-connected component containing Θ(nj) j-sets undergoes a phase transition and the threshold occurs at edge probability, which controls the structure of the component grown in the search process.
Cycles in a Random Graph Near the Critical Point
  • T. Luczak
  • Mathematics
    Random Struct. Algorithms
  • 1991
TLDR
The limit distribution is found of the length of shortest cycle contained in the largest component of G(n, M), as well as of the longest cycle outside it, and it is shown that the probability tending to 1 as n‐→∞ thelength of the shortest cycle in G( n, M) is of the order s2(n)/n.
The longest path in a random graph
A random graph with (1+ε)n/2 edges contains a path of lengthcn. A random directed graph with (1+ε)n edges contains a directed path of lengthcn. This settles a conjecture of Erdõs.
Threshold and Hitting Time for High-Order Connectedness in Random Hypergraphs
TLDR
A hitting time result is deduced for the random hypergraph process –  the hypergraph becomes $j-connected at exactly the moment when the last isolated $j$-set disappears.
A scaling limit for the length of the longest cycle in a sparse random graph
I and J