GKM actions on almost quaternionic manifolds

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 GKM₃ setting the 2-faces of the GKM graph can naturally be divided into quaternionic and complex 2-faces; it turns out that for GKM₃ actions on positive quaternion-K\"ahler manifolds the quaternionic 2-faces are biangles or triangles, and the complex 2-faces triangles or quadrangles. We show purely combinatorially that any abstract GKM₃ 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 H Pⁿ or the Grassmannian Gr₂ (Cⁿ) of complex 2-planes in Cⁿ.