一类优美图

A Class of Graceful Graph

设 u、v是两个固定顶点 ,用 b条内部互不相交且长度皆为 a的道路连接 u、v所得的图用 Pa,b 表示 .K.M.Kathiresan证实 P2 r,2 m- 1 ( r,m皆为任意正整数 )是优美的 ,且猜想 :除了 ( a,b) =( 2 r-1 ,4 m -2 )外 ,所有的 Pa,b都是优美的 .该文第二作者已证实 P2 r- 1 ,2 m- 1 是优美的 ,并且证实了当 r≤ 7时的 P2 r,2 m 也是优美的 .该文证实 r=9时 P2 r,2 m 也是优美的 .

Let u and v be two fixed vertices,connect u and v by means of b internally disjoint paths of length an each,the resulting graphs are denoted by P a,b. K M.Kathiresan has shown that P 2r,2m-1 is graceful and conjectured that P a,b is graceful except when a=2r-1 and b=4m-2. The second author of this paper has shown that P 2r-1,2m-1 and P 2r,2m (r7) are graceful. In this paper, P 2r,2m is proved to be graceful for r=9.