| For a simple graph G=(V,E),if each vertex v V is assigned a non-negative integer f(v)(called labeling of vertex v), and meet three conditions: (l)For v1,v2 V, if v1 v 2,then f(v1) (v2);(2)Max{f(v)|v F} = |E |;(3)For e1,e2 E, if e1 e2, then g(e1) g(e2), and g(e)=[f(u)-f(v)|,e=uv. Then f is called a graceful labeling of G, and G is called a graceful graph.Let u and v be two fixed vertices. We connect u and v by means of "b" internally disjoint paths of length "a" each. The resulting graph is denoted by Pa,b- KM. Kathiresan has shown that P2r,2m-1 is graceful and conjectured that Pa,b is graceful except when (a, b) = (2r-1, 4m- 2) .My instructor Professor Yang YuanSheng has shown that P2r-1,2m-1 and P2r,2m(r 4) are graceful. In this paper, P2r,2m is proved to be graceful for r=5,6,7,9, and then the range for P2r,2m to be graceful is extended to r=7, r=9. |
