Edge correlations in random regular hypergraphs and applications to subgraph testing

TL;DR

This paper investigates correlations in random regular hypergraphs, develops an edge-switching technique to analyze them, and applies these findings to subgraph counting, Hamilton cycle thresholds, and subgraph-freeness testing.

Contribution

It introduces a new edge-switching method for hypergraphs, extending correlation results and applying them to subgraph counts, Hamilton cycles, and property testing.

Findings

Correlations in random regular hypergraphs are limited for a wide density range.

Proved a conjecture on the threshold for $\,\ell$-overlapping Hamilton cycles.

Extended bounds on subgraph-freeness testing from graphs to hypergraphs.

Abstract

Compared to the classical binomial random (hyper)graph model, the study of random regular hypergraphs is made more challenging due to correlations between the occurrence of different edges. We develop an edge-switching technique for hypergraphs which allows us to show that these correlations are limited for a large range of densities. This extends some previous results of Kim, Sudakov and Vu for graphs. From our results we deduce several corollaries on subgraph counts in random $d$-regular hypergraphs. We also prove a conjecture of Dudek, Frieze, Ruci\'nski and \v{S}ileikis on the threshold for the existence of an $\ell$-overlapping Hamilton cycle in a random $d$-regular $r$-graph. Moreover, we apply our results to prove bounds on the query complexity of testing subgraph-freeness. The problem of testing subgraph-freeness in the general graphs model was first studied by Alon, Kaufman, Krivelevich and Ron, who obtained several bounds on the query complexity of testing triangle-freeness. We extend some of these previous results beyond the triangle setting and to the hypergraph setting.