On the triangle space of a random graph

TL;DR

This paper proves that in a random graph, the triangles typically generate the entire cycle space when each edge is part of a triangle, confirming a special case of Kahle's conjecture.

Contribution

It establishes the conditions under which the triangle subgraph spans the cycle space in random graphs, providing two proofs and a new stability theorem for triangle-free subgraphs.

Findings

Triangles span the cycle space when p is above about √(1.5 ln n / n)

High probability of triangle coverage in the cycle space under certain p

New simple proof of a stability theorem for triangle-free subgraphs

Abstract

Settling a first case of a conjecture of M. Kahle on the homology of the clique complex of the random graph $G=G_{n,p}$, we show, roughly speaking, that (with high probability) the triangles of $G$ span its cycle space whenever each of its edges lies in a triangle (which happens (w.h.p.) when $p$ is at least about $\sqrt{(3/2)\ln n/n}$, and not below this unless $p$ is very small.) We give two related proofs of this statement, together with a relatively simple proof of a fundamental "stability" theorem for triangle-free subgraphs of $G_{n,p}$, originally due to Kohayakawa, \L uczak and R\"odl, that underlies the first of our proofs.