Summary: We consider 6-regular plane graphs whose faces have size \(1, 2\) or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to \(53\) vertices. Subsequently, in Section 3 we enumerate all possible symmetry groups of the spheres that showed up. In Section 4 we introduce a new Goldberg-Coxeter construction that takes a 6-regular plane graph \(G_0\), two integers \(k\) and \(l\) and returns two 6-regular plane graphs.
Then in the final section, we consider the notions of zigzags and central circuits for the considered graphs. We introduced the notions of tightness and weak tightness for them and prove an upper bound on the number of zigzags and central circuits of such tight graphs. We also classify the tight and weakly tight graphs with simple zigzags or central circuits.