The word problem and the Aharoni-Berger-Ziv conjecture on the connectivity of independence complexes

TL;DR

This paper challenges a conjecture relating a recursively defined number to the connectivity of independence complexes in graphs, providing a short disproof using recursion theory.

Contribution

It offers a short disproof of the conjecture that the lower bound for connectivity is always optimal, using recursion theory techniques.

Findings

The conjecture by Aharoni, Berger, and Ziv is false.

Recursion theory can be applied to problems in topological graph theory.

The paper provides a counterexample to the conjecture.

Abstract

For each finite simple graph $G$, Aharoni, Berger and Ziv consider a recursively defined number $\psi (G) \in \mathbb{Z}\cup \{+ \infty \}$ which gives a lower bound for the topological connectivity of the independence complex $I_G$. They conjecture that this bound is optimal for every graph. We use a result of recursion theory to give a short disproof of this claim.