Right-angled Artin groups and full subgraphs of graphs

Abstract

For a finite graph [Formula: see text], let [Formula: see text] be the right-angled Artin group defined by the complement graph of [Formula: see text]. We show that, for any linear forest [Formula: see text] and any finite graph [Formula: see text], [Formula: see text] can be embedded into [Formula: see text] if and only if [Formula: see text] can be realized as a full subgraph of [Formula: see text]. We also prove that if we drop the assumption that [Formula: see text] is a linear forest, then the above assertion does not hold, namely, for any finite graph [Formula: see text], which is not a linear forest, there exists a finite graph [Formula: see text] such that [Formula: see text] can be embedded into [Formula: see text], though [Formula: see text] cannot be embedded into [Formula: see text] as a full subgraph.