一类图的边数

The numbers of edge of one class graphs

设G是具有n个顶点的图，ai是G中长为i的圈的个数，h（n，m，r）为满足对于所有的i，ai≤m且当i＜r时，ai=0的图的最大可能边数。边数为h（n，m，r）并满足对于所有的i，ai≤m且当i＜r时，ai=0的图G全体记为H，本文给出了h（n，m，r）的上、下界并证明当n≥nｏ时，H中的作有图G都不包含n圈。对于Erdos问题，本文绘出了f（n）的新上界。

Let G be a graph with n vertices,ai be the numbers of cycle of len gth i in G, ε (G) be the numbers of edge of G, let h(n, m,r)=max{ε (G) ｜ ai≤m for all i and ai=o for all i<r}. f(n) =h(n, 1, 1). In 1975, Erdos raised the problem of determine f(n). In this paper we give the upper bound and lower bound of f(n)<n+1 h(n, m, r)and give the new upper bound of f(n) : f(n) < for all n > ((2m+3)/4) e2m.