Loose Cores and Cycles in Random Hypergraphs

  title={Loose Cores and Cycles in Random Hypergraphs},
  author={Oliver Cooley and Mihyun Kang and Julian Zalla},
Inspired by the study of loose cycles in hypergraphs, we define the loose core in hypergraphs as a structure which mirrors the close relationship between cycles and 2-cores in graphs. We prove that in the r-uniform binomial random hypergraph H(n, p), the order of the loose core undergoes a phase transition at a certain critical threshold and determine this order, as well as the number of edges, asymptotically in the subcritical and supercritical regimes. Our main tool is an algorithm called… 
1 Citations
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We formulate some simple conditions under which a Markov chain may be approximated by the solution to a differential equation, with quantifiable error probabilities. The role of a choice of


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    Random Struct. Algorithms
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We study random r‐uniform n vertex hypergraphs with fixed degree sequence d = (d1…,dn), maximum degree Δ = o(n1/24) and total degree θn, where θ is bounded. We give the size, number of edges and
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The degree of a vertex in a hypergraph is defined as the number of edges incident to it. In this paper we study the $k$-core, defined as the maximal induced subhypergraph of minimum degree $k$, of
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    Combinatorics, Probability and Computing
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It is shown that the fluctuations around the deterministic limit converge to a Gaussian law above and near the threshold, and to a non-normal law at the threshold.
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This work recovers the result by Pittel, Spencer and Wormald on the existence and size of a k-core in G(n,p) and G( n,m), based on the properties of empirical distributions of independent random variables, and leads to simple proofs.
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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
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It is shown that the same method applies to the analogous model of random k ‐uniform hypergraphs, establishing asymptotic normality throughout the (sparse) supercritical regime.
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    Random Struct. Algorithms
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