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Largest Components in Random Hypergraphs


In this paper we consider j-tuple-connected components in random k-uniform hypergraphs (the j-tuple-connectedness relation can be defined by letting two j-sets be connected if they lie in a common edge and considering the transitive closure; the case j = 1 corresponds to the common notion of vertex-connectedness). We show that the existence of a j-tuple-connected component containing Θ(nj) j-sets undergoes a phase transition and show that the threshold occurs at edge probability

Our proof extends the recent short proof for the graph case by Krivelevich and Sudakov, which makes use of a depth-first search to reveal the edges of a random graph.

Our main original contribution is a bounded degree lemma, which controls the structure of the component grown in the search process.

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[1] Aronshtam, L. and Linial, N. (2015) When does the top homology of a random simplicial complex vanish? Random Struct. Alg. 46 2635.
[2] Aronshtam, L. and Linial, N. (2016) The threshold for d-collapsibility in random complexes. Random Struct. Alg. 48 260269.
[3] Aronshtam, L., Linial, N., Łuczak, T. and Meshulam, R. (2013) Collapsibility and vanishing of top homology in random simplicial complexes. Discrete Comput. Geom. 49, no. 2, 317334.
[4] Behrisch, M., Coja-Oghlan, A. and Kang, M. (2010) The order of the giant component of random hypergraphs. Random Struct. Alg. 36 149184.
[5] Behrisch, M., Coja-Oghlan, A. and Kang, M. (2014) Local limit theorems for the giant component of random hypergraphs. Combin. Probab. Comput. 23 331366.
[6] Bohman, T. (2009) The triangle-free process. Adv. Math. 221 16531677.
[7] Bollobás, B. (1984) The evolution of random graphs. Trans. Amer. Math. Soc. 286 257274.
[8] Bollobás, B. (2001) Random Graphs, second edition, Cambridge University Press.
[9] Bollobás, B. and Riordan, O. (2012) Asymptotic normality of the size of the giant component in a random hypergraph. Random Struct. Alg. 41 441450.
[10] Cooley, O., Kang, M. and Koch, K. The size of the giant component in random hypergraphs. Random Struct. Alg., DOI: 10.1002/rsa.20761.
[11] Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 1761.
[12] Janson, S., Łuczak, T. and A. Ruciński, A. (2000) Random Graphs, Wiley.
[13] Karoński, M. and Łuczak, T. (2002) The phase transition in a random hypergraph. J. Comput. Appl. Math. 142 125135.
[14] Krivelevich, M. and Sudakov, B. (2013) The phase transition in random graphs: A simple proof. Random Struct. Alg. 43 131138.
[15] Linial, N. and Meshulam, R. (2006) Homological connectivity of random 2-complexes. Combinatorica 26 475487.
[16] Lu, L. and Peng, X. High-order phase transition in random hypergraphs. arXiv:1409.1174
[17] Łuczak, T. (1990) Component behavior near the critical point of the random graph process. Random Struct. Alg. 1 287310.
[18] Molloy, M. (2005) Cores in random hypergraphs and boolean formulas. Random Struct. Alg. 27 124135.
[19] Nachmias, A. and Peres, Y. (2010) The critical random graph, with martingales. Israel J. Math. 176 2941.
[20] Ravelomanana, and Rijamamy, (2006) Creation and growth of components in a random hypergraph process. In COCOON 2006: Computing and Combinatorics, Springer, pp. 350359.
[21] Schmidt-Pruzan, J. and E. Shamir, E. (1985) Component structure in the evolution of random hypergraphs. Combinatorica 5 8194.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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