The homotopy type of the independence complex of graphs with no induced cycles of length divisible by $3$

TL;DR

This paper proves a conjecture that the independence complex of certain graphs without cycles of length divisible by 3 is either contractible or a sphere, confirming a topological classification and Betti number bound.

Contribution

It establishes the homotopy type of independence complexes for graphs with no induced cycles of length divisible by 3, confirming a conjecture and strengthening previous results.

Findings

Independence complex is either contractible or homotopy equivalent to a sphere.

Total Betti number of the independence complex is at most 1.

Verifies a conjecture relating to the topology of such graphs.

Abstract

We prove Engstr\"{o}m's conjecture that the independence complex of graphs with no induced cycle of length divisible by $3$ is either contractible or homotopy equivalent to a sphere. Our result strengthens a result by Zhang and Wu, verifying a conjecture of Kalai and Meshulam which states that the total Betti number of the independence complex of such a graph is at most $1$. A weaker conjecture was proved earlier by Chudnovsky, Scott, Seymour, and Spirkl, who showed that in such a graph, the number of independent sets of even size minus the number of independent sets of odd size has values $0$, $1$, or $-1$.