Graphs of 2-torus actions

It has been known that an effective smooth $({\Bbb Z}_2)^k$-action on a smooth connected closed manifold $M^n$ fixing a finite set can be associated to a $({\Bbb Z}_2)^k$-colored regular graph. In this paper, we consider abstract graphs $(\Gamma,\alpha)$ of $({\Bbb Z}_2)^k$-actions, called abstract 1-skeletons. We study when an abstract 1-skeleton is a colored graph of some $({\Bbb Z}_2)^k$-action. We also study the existence of faces of an abstract 1-skeleton (note that faces often have certain geometric meanings if an abstract 1-skeleton is a colored graph of some $({\Bbb Z}_2)^k$-action).