图P_(2r，2m)的优美标号

THE GRACEFUL LABELINGS OF P_(2r,2m)

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所得图用P_(a,b)表示．K．M．Kathiresan证明：P_(2r,2m-1)(r,m皆为任意正整数)是优美的,且猜想,除了(a,b)=(2r+1,4s+2)外,所有的P_(a,b)都是优美的．杨元生已证明P_(2r+1,2m-1)是优美的,并且证明了,当r=1,2,3,4,5,6,7时,P_(2r,2m)也是优美的．作者证明：r为任意奇数时,P_(2r,2m)也是优美的．

Let u and v be two fixed vertices. Connect u and v by b internally disjoint paths of length a and denote the resulting graphs by Pa,b. K. M. Kathiresan shows that P2r,2m-1 is graceful and conjectures that Pa,b is graceful except when （a, b） = （2r ＋ 1, 4s ＋ 2）. Y. S. Yang shows that P2r＋1,2m-i and P2r,2m（r = 1, 2,..., 7） are graceful. In this paper, P2r,2m is proved to be graceful when r is an odd number.