**No.**
976

**Title:**

- Face-regular 3-valent two-faced spheres and tori

**Author(s): **

- Mathieu, DUTOUR SIKIRIC (ISM)

- Michel, DEZA (ENS)

**Key words: **

- two-faced polyhedra, plane graphs, toroidal maps, computer enumeration.

**Abstract: **

- Any
*(3,q)*-map, which is*3R0*, has*4 <= q <= 12*. It is strongly face-regular for*q=4,5*(*Prism(3)*and*Barrel(3)*only) and*q=11,12*(only tori, unique for*q=12*). All such weakly face-regular maps are infinities of polyhedra and tori for each*7 <= q <= 10*and (characterized) infinity of*(3,6)*-polyhedra. - Weakly face-regular
*(5,q)*-polyhedra*5Rj*exist for*j=3, 6 <= q <= 10*, and*j=2, q >= 8*. - The following general conjecture: the number
of
*(p,q)*-polyhedra, which are*qRj*, is infinite if and only the corresponding torus exist. - If a
*(p,q)*-polyhedron is*qRj*, then*j <= 5*. The number of*(5,q)*-polyhedra is infinite if and only if*q >= 12,10,8,7,7,7*for*j=0,1,2,3,4,5*, respectively (except some undecided cases for*j=0,3,5*). - The number of
*(4,q)*-polyhedra*qRj*is finite (all are classified) for*j <= 3*. For*j=4*, they are classified (and their number is infinite) if*q=8*; we conjecture infiniteness if and only if*q <= 8*. - For many classes
*qRj*, the number of possibilities is very large. However, for the critical value of*q*, starting at which an infinity of graphs occurs, we are often able to describe the structure and obtain a complete classification.

Call

We say that a

All strictly face-regular

Here we address the characterization of all