A pr 2 00 1 Clusters of Cycles

A cluster of cycles (or (r, q)-polycycle) is a simple planar 2–connected finite or countable graph G of girth r and maximal vertex-degree q, which admits an (r, q)-polycyclic realization P (G) on the plane. An (r, q)-polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q, (ii) all interior faces (denote their number by pr) are combinatorial r-gons, (iii) all vertices, edges and interior faces form a cell-complex. An example of (r, q)-polycycle is the skeleton of (r), i.e. of the q-valent partition of the sphere, Euclidean plane or hyperbolic plane by regular r-gons. Call spheric pairs (r, q) = (3, 3), (4, 3), (3, 4), (5, 3), (3, 5). Only for those five pairs, P ((r)) is (r) without exterior face; otherwise, P ((r)) = (r). Here we give a compact survey of results on (r, q)-polycycles. We start with the following general results for any (r, q)-polycycle G: (i) P (G) is unique, except of (easy) case when G is the skeleton of one of 5 Platonic polyhedra; (ii) P (G) admits a cell-homomorphism f into (r); (iii) a polynomial criterion to decide if given finite graph is a polycycle, is presented. Call a polycycle proper if it is a partial subgraph of (r) and a helicene, otherwise. In [18] all proper spheric polycycles are given. An (r, q)-helicene exists if and only if pr > (q − 2)(r − 1) and (r, q) 6= (3, 3). We list the (4, 3)-, (3, 4)-helicenes and the number of (5, 3)-, (3, 5)-helicenes for first interesting pr. Any outerplanar (r, q)polycycle G is a proper (r, 2q−2)-polycycle and its projection f(P (G)) into (r2q−2) is convex. Any outerplanar (3, q)-polycycle G is a proper (3, q + 2)-polycycle. The symmetry group Aut(G) (equal to Aut(P (G)), except of Platonic case) of an (r, q)-polycycle G is a subgroup of Aut((r)) if it is proper and an extension of Aut(f(P (G))), otherwise. Aut(G) consists only of rotations and mirrors if G is finite; so its order divides one of the numbers 2r, 4 or 2q. Almost all polycycles G have trivial AutG. first author acknowledges financial support of Mathematical Institute at Ohio State University second author acknowledges financial support of the Russian Foundation for Fundamental Research (grant 99-01-00010)