P_(2r,4m+2)图(r=8,10)的优美性

Gracefulness of Graph P_(2r,4m+2)(r=8,10)

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P2r,2m-1(r,m皆为任意正整数)是优美的,且猜想:除了(a,b)=(2r-1,4m-2)外,所有的Pa,b都是优美的.杨元生教授已证实P2r-1,2m-1是优美的,并且证实了当r=1,2,3,4,5,6,7,9时P2r,2m也是优美的.该文证实当r=8,10时P2r,4m+2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 a each,the resulting graphs is denoted by Pa,b.K.M.Kathiresan has shown that P2r,2m-1 is graceful and conjectured that Pa,b is graceful except when（a,b）=（2r-1,4m-2）.Prof.YANG Yuansheng has shown that P2r-1,2m-1 and P2r,2m（r=1,2,3,4,5,6,7,9） are graceful.In this paper,P2r,4m＋2 is proved to be graceful for r=8,10.