The Graceful Labelings of P2r,2m
图P2r,2m的优美标号

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 $P_{a, b}$.K. M. Kathiresan shows that $P_{2r, 2m-1}$ is graceful and conjectures that $P_{a, b}$ is graceful except when $(a,b)=(2r+1,4s+2)$. Y. S. Yang shows that $P_{2r+1,2m-1}$ and $P_{2r, 2m}(r=1, 2, \cdots, 7)$ are graceful. In this paper, $P_{2r, 2m}$ is proved to be graceful when $r$ is an odd number.