Asphericity of groups defined by graphs

A graph $\mathrm{\Gamma }$ labelled by a set $S$ defines a group $G(\mathrm{\Gamma })$ whose generators are the set of labels $S$ and whose relations are all words which can be read on closed paths of this graph. We introduce the notion of aspherical graph and prove that such a graph defines an aspherical group presentation. This result generalizes a theorem of Dominik Gruber on graphs satisfying graphical $C(6)$-condition and also allows to get new graphical conditions of asphericity analogous to some classical conditions.