Spheres and tori that are bR

In this chapter, we obtain full classification of ( 4, b), 3)-maps that are bRj. For ( 5, b), 3)-maps we have only existence results. 

The (4, 3)-polycycles used in theorem below are defined in Section 4.2. 

Every vertex, which is incident to three fi-gonal faces, corresponds to a 3-gonal face of b(G). Every (4,3)-polycycle 4, 3 — v also corresponds to a 3-gonal face. Every (4,3)-polycycle 4,3 — e corresponds to a 2-gonal face. On the other hand, all P2 x Pk correspond to 4-gonal faces. A 3-valent map, whose faces have gonality at most four, does not exist on the torus and, clearly, has at most 8 vertices on the sphere by Euler Formula 1.1. 

G is a 3-valent sphere with faces of gonality at most four. There are exactly five such maps Tetrahedron, Bundle3, Prism2, Prism2, and Cube. 

If G is Tetrahedron, then all its faces are 3-gonal hence, they all correspond to (4, 3)-polycycles 4, 3 — v or to vertices. Clearly, in order for a face to be 6-gonal, they should be all 4, 3 — v or all vertices. This corresponds to strictly face-regular Nr. 35 or to a Tetrahedron. 

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