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:

Call two-faced map and, specifically, (p,q)-map a 3-valent map (on sphere or torus) with only p- and q-gonal faces (at least one each), for given integers 3<=p < q; so, 3<= p<= 5. Two-faced maps (especially, (5,6)-polyhedra, called fullerenes) are prominent molecular models in Chemistry.

We say that a (p,q)-map is pRi if any p-gon has the same number i of p-gonal neighbors; it is qRj if each q-gon has the same number j of q-gonal neighbors. Call a (p,q)-map strictly face-regular if it is both, pRi and qRj, for some i, j; call it weakly face-regular, if it is only pRi or qRj.

All strictly face-regular (p,q)-polyhedra are ([BrDe99], [De02]) Prism(m), Barrel(m) and 55 sporadic polyhedra. By Barrel(m) we denote 4m-vertex (5,m)-polyhedron, consisting of two m-gons separated by two m-rings of 5-gons. All 23 parameter-sets (p,q;i,j) for strictly face-regular (p,q)-tori are found ([De02]); the number of minimal tori is one for 7 of them and an infinity for 16 others.

Here we address the characterization of all weakly face-regular (p,q)-maps on sphere or torus. Examples of obtained results are:
• 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.
We used large computations, variations of Euler formula, analysis of coronae of faces, the decomposition of (p,q)-maps into elementary (p,3)-polycycles and many ad hoc arguments.