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arXiv · 2003.14143

Longest paths in random hypergraphs

Oliver Cooley·Frederik Garbe·Eng Keat Hng·Mihyun Kang·Nicolás Sanhueza-Matamala·Julian Zalla
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Abstract

Given integers $k,j$ with $1\le j \le k-1$, we consider the length of the longest $j$-tight path in the binomial random $k$-uniform hypergraph $H^k(n,p)$. We show that this length undergoes a phase transition from logarithmic length to linear and determine the critical threshold, as well as proving upper and lower bounds on the length in the subcritical and supercritical ranges. In particular, for the supercritical case we introduce the `Pathfinder' algorithm, a depth-first search algorithm which discovers $j$-tight paths in a $k$-uniform hypergraph. We prove that, in the supercritical case, with high probability this algorithm will find a long $j$-tight path.

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Oliver Cooley, Frederik Garbe, Eng Keat Hng, Mihyun Kang, Nicolás Sanhueza-Matamala, Julian Zalla. 2020-03-31. Longest paths in random hypergraphs. https://arxiv.org/abs/2003.14143

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