Abstract
Denote by an ℓ-component a connected b-uniform hypergraph with k edges and k(b–1) – ℓ vertices. We prove that the expected number of creations of ℓ-component during a random hypergraph process tends to 1 as ℓ and b tend to ∞ with the total number of vertices n such that \(\ell = o\left( \sqrt[3]{\frac{n}{b}} \right)\). Under the same conditions, we also show that the expected number of vertices that ever belong to an ℓ-component is approximately 121/3 (b–1)1/3 ℓ1/3 n 2/3. As an immediate consequence, it follows that with high probability the largest ℓ-component during the process is of size O( (b–1)1/3 ℓ1/3 n 2/3 ). Our results give insight about the size of giant components inside the phase transition of random hypergraphs.
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Ravelomanana, V., Rijamamy, A.L. (2006). Creation and Growth of Components in a Random Hypergraph Process. In: Chen, D.Z., Lee, D.T. (eds) Computing and Combinatorics. COCOON 2006. Lecture Notes in Computer Science, vol 4112. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11809678_37
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DOI: https://doi.org/10.1007/11809678_37
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