We introduce quaternionic structures on abstract GKM graphs, as the combinatorial counterpart of almost quaternionic structures left invariant by a torus action of GKM type. In the \(\hbox {GKM}_3\) setting the 2-faces of the GKM graph can naturally be divided into quaternionic and complex 2-faces; it turns out that for \(\hbox {GKM}_3\) actions on positive quaternion-Kähler manifolds the quaternionic 2-faces are biangles or triangles, and the complex 2-faces triangles or quadrangles. We show purely combinatorially that any abstract \(\hbox {GKM}_3\) graph with quaternionic structure satisfying this restriction on the 2-faces of the GKM graph is that of a torus action on quaternionic projective space \({\mathbb {H}}P^n\) or the Grassmannian \({\textrm{Gr}}_2({\mathbb {C}}^n)\) of complex 2-planes in \({\mathbb {C}}^n\) .
