周期簇
Clusters of Cycles
摘要：
一个{ \it循环簇}（或{ \it $(r,q)$ -polycycle}）是围长$r$和最大顶点度$q$的简单平面2-连通有限或可数图$G$，它在平面上允许{ \it $(r,q)$ -polycycle实现}，用$P(G)$表示它，即使得：（i）所有的内顶点都是$q$度，（ii）所有的内面（用$p_r$表示它们的数目）都是组合$r$-MQ，并且（由（i）、（ii）暗示）（iii）所有的顶点、边和内面形成胞腔复形。$(r,q)$ -多环的一个例子是$(r^q)$的骨架，即球形$S^2$、欧几里得平面$R^2$或双曲平面$H^2$被规则$r$ -环的$q$-配分.调用{ \it sphere}对$(r,q)=(3,3),(3,4),(4,3),(3,5),(5,3)$ ;对于这五对，$P(r^q)$是没有外表面的$(r^q)$;否则为$P(r^q)=(r^q)$。我们在这里给一个紧凑的调查结果$(r,q)$ -多环。
英文摘要：
A {\it cluster of cycles} (or {\it $(r,q)$-polycycle}) is a simple planar 2--co nnected finite or countable graph $G$ of girth $r$ and maximal vertex-degree $q$, which admits {\it $(r,q)$-polycyclic realization} on the plane, denote it by $P(G)$, i.e. such that: (i) all interior vertices are of degree $q$, (ii) all interior faces (denote their number by $p_r$) are combinatorial $r$-gons and (implied by (i), (ii)) (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 $S^2$, Euclidean plane $R^2$ or hyperbolic plane $H^2$ by regular $r$-gons. Call {\it spheric} pairs $(r,q)=(3,3),(3,4),(4,3),(3,5),(5,3)$; for those five pairs $P(r^q)$ is $(r^q)$ without the exterior face; otherwise $P(r^q)=(r^q)$. We give here a compact survey of results on $(r,q)$-polycycles.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
