Enumerating independent sets in Abelian Cayley graphs

TL;DR

This paper establishes an asymptotically tight upper bound on the number of independent sets in connected Abelian Cayley graphs with degree logarithmic in the group order, using advanced combinatorial methods.

Contribution

It introduces a novel application of the graph container method combined with additive combinatorics to bound independent sets in Abelian Cayley graphs.

Findings

Bound is tight for bipartite graphs

Uses Sapozhenko's graph container method

Employs Plünnecke-Rusza-Petridis inequality

Abstract

We show that any connected Cayley graph $\Gamma$ on an Abelian group of order $2n$ and degree $\tilde{\Omega}(\log n)$ has at most $2^{n+1}(1 + o(1))$ independent sets. This bound is tight up to to the $o(1)$ term when $\Gamma$ is bipartite. Our proof is based on Sapozhenko's graph container method and uses the Pl\"{u}nnecke-Rusza-Petridis inequality from additive combinatorics.