Longest paths in random hypergraphs

Abstract

Given integers k, j with 1 ≤ j ≤ 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.

Paths, cycles and sprinkling in random hypergraphs

“…Note that it is trivially true that L P ≥ L C − O (1), where L P denotes the length of the longest j-tight path in H k (n, p). Therefore as a corollary we also obtain a lower bound on L P which generalises the one in [3].…”

“…Let H k (n, p) denote the k-uniform binomial random hypergraph, in which each k-set of vertices forms an edge with probability p independently. The analogue of the result of Ajtai, Komlós and Szemerédi showing a threshold for the existence of a j-tight path of linear length in H k (n, p) was proved by the author together with Garbe, Hng, Kang, Sanhueza-Matamala and Zalla [3] for all k and j. In contrast to the graph case, in general the threshold is not the same as the threshold for a giant j-tuple component (which was determined in [7]).…”

“…. The results of [3] show that p 0 is a threshold for the existence of a j-tight path of linear length in H k (n, p). Furthermore, in the case when p = (1 + ε)p 0 for some constant ε > 0, upper and lower bounds on the length of the longest j-tight path were proved.…”

“…The strategy used was to prove an upper bound on the length of the longest loose cycle which transfers to an upper bound for loose paths using a standard sprinkling argument, just as has been often observed for graphs. Similarly, sprinkling can also be used to extend the lower bound on loose paths from [3] to an asymptotically identical lower bound for loose cycles.…”

“…We thus provide a lower bound on the length of the longest j-tight cycle. Along the way, we generalise the lower bound for j-tight paths given in [3] to be applicable for a larger range of p.…”