P_(3,4m)的优美性

The Gracefulness of P_(3,4m)

设u、v是两个固定顶点,用b条内部互不相交且长度均为a的道路连接u、v所得到的图用Pa,b表示。Kathiresan证实P2r,2m-1(r,m均为任意正整数)是优美的,且猜想:除了(a,b)=(2r+1,4s+2)外,所有的Pa,b都是优美的。杨元生已证实P2r+1,2m-1是优美的。本文证明P3,4m是优美图,从而当a=3时Kathiresan猜想成立。

Let u and v be two fixed vertices. Connect u and v by b internally disjoint paths of length a , the resulting graphs denoted by Pa,b. Kathiresan showed that P2r.2m-1 is graceful and conjectured that Pa,b is graceful except for （ a, b） = （2r ＋ 1,4s ＋ 2） .Yang showed that P2r＋1,2m-1 are graceful, In this paper , P3,4m is proved to be graceful. So we prove the conjecture for a = 3.