On Betti numbers of flag complexes with forbidden induced subgraphs

TL;DR

This paper investigates the asymptotic growth of Betti numbers in clique complexes of graphs avoiding a fixed induced subgraph, revealing exactly five possible exponential growth rates and providing bounds for specific cases.

Contribution

It establishes a classification of the exponential growth rates of Betti numbers for graphs with forbidden induced subgraphs, including the case of 4-cycles.

Findings

Five distinct exponential growth limits for Betti numbers are identified.

For H as a 4-cycle, the growth rate limit is exactly 1.

A slightly superpolynomial upper bound is proved for the 4-cycle case.

Abstract

We analyze the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H. In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained. For the interesting case where H is the 4-cycle, the above limit is 1, and we prove a slightly superpolynomial upper bound.