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 \tt 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.
Original language | English |
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Pages (from-to) | 2430-2458 |
Journal | SIAM Journal on Discrete Mathematics |
Volume | 35 |
Issue number | 4 |
DOIs | |
Publication status | Published - 2021 |
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
Fields of Expertise
- Information, Communication & Computing