Clusters of cycles

Abstract

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 p r ) 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 q ), 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 q )) is ( r q ) without exterior face; otherwise, P(( r q ))=( r q ). 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 the five Platonic polyhedra; (ii) P( G) admits a cell-homomorphism f into ( r q ); (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 q ) and a helicene, otherwise. In [ARS Comb. A 29 (1990) 5], all proper spheric polycycles are given. An ( r, q)-helicene exists if and only if p r >( q−2)( r−1) and ( r, q)≠(3,3). We list the (4,3)-, (3,4)-helicenes and the number of (5,3)-, (3,5)-helicenes for first interesting p r . Any outerplanar ( r, q)-polycycle G is a proper ( r,2 q−2)-polycycle and its projection f( P( G)) into ( r 2 q−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 q )) 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 2 r, 4 or 2 q. Almost all polycycles G have trivial AutG. Call a polycycle G isotoxal (or isogonal, or isohedral) if AutG is transitive on edges (or vertices, or interior faces); use notation IT (or IG, or IH), for short. Only r-gons and non-spheric ( r q ) are isotoxal. Let T ∗(l,m,n) denote Coxeter’s triangle group of a triangle on S 2, E 2 or H 2 with angles π/ l, π/ m, π/ n and let T( l, m, n) denote its subgroup of index 2, excluding motions of 2nd kind. We list all IG- or IH-polycycles for spheric ( r, q) and construct many examples of IH-polycycles for general case (with AutG being above two groups for some parameters, including strip and modular groups). Any IG-, but not IT-polycycle is infinite, outerplanar and with same vertex-degree, we present two IG-, but not IH-polycycles with ( r, q)=(3,5),(4,4) and AutG= T(2,3,∞)∼ PSL(2, Z), T ∗(2,4,∞) . Any IH-polycycle has the same number of boundary edges for each its r-gon. For any r≥5, there exists a continuum of quasi-IH-polycycles, i.e. not isohedral, but all r-gons have the same 1-corona. On two notions of extremal polycycles: 1. We found for the spheric ( r, q) the maximal number n int of interior points for an ( r, q)-polycycle with given p r ; in general case, ( p r / q)≤ n int<( rp r / q) if any r-gon contains an interior point. 2. All non-extendible ( r, q)-polycycles (i.e. not a proper subgraph of another ( r, q)-polycycle) are ( r q ), four special ones, (possibly, but we conjecture their non-existence) some other finite (3,5)-polycycles, and, for any ( r, q)≠(3,3),(3,4),(4,3), a continuum of infinite ones. On isometric embedding of polycycles into hypercubes Q m , half-hypercubes 1 2 Q m and, if infinite, into cubic lattices Z m , 1 2 Z m : for ( r, q)≠(5,3),(3,5), there are exactly three non-embeddable polycycles (including (4 3)− e, (3 4)− e); all non-embeddable (5,3)-polycycles are characterized by two forbidden sub-polycycles with p 5=6.