Spanning structures and universality in sparse hypergraphs

TL;DR

This paper investigates spanning structures and universality in sparse random hypergraphs, adapting known results from random graphs to hypergraphs and establishing conditions for universality and sparse constructions.

Contribution

It extends Riordan's spanning subgraph results to hypergraphs and analyzes universality thresholds, providing new sparse hypergraph constructions.

Findings

Sufficient conditions for spanning structures in hypergraphs.

Universality holds for hypergraphs with edge probability p=ω((ln n / n)^{1/Δ}).

Sparse universal hypergraphs can be constructed using known graph methods.

Abstract

In this paper the problem of finding various spanning structures in random hypergraphs is studied. We notice that a general result of Riordan [Spanning subgraphs of random graphs, Combinatorics, Probability & Computing 9 (2000), no. 2, 125-148] can be adapted from random graphs to random $r$-uniform hypergaphs and provide sufficient conditions when a random $r$-uniform hypergraph $\mathcal{H}^{(r)}(n,p)$ contains a given spanning structure a.a.s. We also discuss several spanning structures such as cube-hypergraphs, lattices, spheres and Hamilton cycles in hypergraphs. Moreover, we study universality, i.e. when does an $r$-uniform hypergraph contain any hypergraph on $n$ vertices and with maximum vertex degree bounded by $\Delta$? For $\mathcal{H}^{(r)}(n,p)$ it is shown that this holds for $p= \omega \left((\ln n/n)^{1/\Delta}\right)$ a.a.s. by combining approaches taken by Dellamonica, Kohayakawa, R\"odl and Ruci\'nski [An improved upper bound on the density of universal random graphs, Random Structures Algorithms 46 (2015), no. 2, 274-299] and of Ferber, Nenadov and Peter [Universality of random graphs and rainbow embedding, Random Structures Algorithms, to appear]. Furthermore it is shown that the random graph $G(n,p)$ for appropriate $p$ and explicit constructions of universal graphs due to Alon, Capalbo, Kohayakawa, R\"odl, Ruci\'nski and Szemer\'edi and Alon and Capalbo yield constructions of universal hypergraphs that are sparser than the random hypergraph $\mathcal{H}^{(r)}(n,p)$ with $p= \omega \left((\ln n/n)^{1/\Delta}\right)$.