Asphericity of Groups Defined by Graphs

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