Collapsibility of random clique complexes

Abstract

We prove a sufficient condition for a finite clique complex to collapse to a k-dimensional complex, and use this to exhibit thresholds for (k+1)-collapsibility in a sparse random clique complex. In particular, if every strongly connected, pure (k+1)-dimensional subcomplex of a clique complex X has a vertex of degree at most 2k+1, then X is (k+1)-collapsible. In the random model X(n,p) of clique complexes of an Erdős–Rényi random graph G(n,p), we then show that for any fixed k≥0, if p=n−α for fixed α>1/(k+1), then a clique complex X=distX(n,p) is (k+1)-collapsible with high probability.2