非广义多边形路的2连通图的圈数

The Number of Cycles in a 2 Connected Graph Being Not a Generalized Polygon Path

若Ｇ中一条路Ｐ的每个内点ｖ 都有ｄＧ（ｖ）＝ ２，则称Ｐ为Ｇ的简单路⒀一个２ 连通可平面图Ｇ称为广义多边形路，如果用下述方法得到的图Ｇ是路：对应于Ｇ的每个内部面ｆ （Ｇ是Ｇ的平图）有一个Ｇ的顶点ｆ，Ｇ的两个顶点ｆ和ｇ在Ｇ中相邻当且仅当Ｇ中相应的两个内部面的边界交于一条Ｇ的简单路⒀令ｊ＝ ｜Ｅ（Ｇ）｜－ ｜Ｖ（Ｇ）｜和ｍ （Ｇ）为Ｇ的含圈数⒀论文证明了下述结果：设Ｇ是非广义多边形路的２ 连通图，则ｍ （Ｇ）≥ｊ２＋ ５ｊ２ －

A path P in G is called a simple path of G if, for each interior vertex v of P, d G(v)=2. A 2 connected planar graph G is called a generalized polygon path if G * formed by the following method is a path: corresponding to each interior face f of  ( is a plane graph of G), there is a vertex f * of G *; two vertices f * and g * are adjacent in G * if and only if the boundaries of the corresponding interior faces of  intesect a simple path of 。 Let j=｜E(G)｜-｜V(G)｜ and m(G) be the number of cycles in G. We prove the following result: Let G be a 2 connected graph being not a generalized polygon path, then m(G)≥j 2+5j2-1.