Pairwise disjoint maximal cliques in random graphs and sequential motion planning on random right angled Artin groups

TL;DR

This paper studies the structure of random graphs and applies the findings to analyze the topological complexity of spaces related to random right angled Artin groups, extending known properties of clique numbers.

Contribution

It extends the understanding of clique structures in random graphs and applies these results to the topological analysis of random graph groups.

Findings

Random graphs almost surely contain many disjoint complete subgraphs of size r(n,p).

The paper provides an asymptotic description of higher topological complexities of associated spaces.

Results connect graph theory with topological properties of random groups.

Abstract

The clique number of a random graph in the Erdos-Renyi model G(n,p) yields a random variable which is known to be asymptotically (as n tends to infinity) almost surely within one of an explicit logarithmic (on n) function r(n,p). We extend this fact by showing that random graphs have, asymptotically almost surely, arbitrarily many pairwise disjoint complete subgraphs with as many vertices as r(n,p). The result is motivated by and applied to the sequential motion planning problem on random right angled Artin groups. Indeed, we give an asymptotical description of all the higher topological complexities of Eilenberg-MacLane spaces associated to random graph groups.