关于B(m,n,p)和B(m,n)的1-优美性

The 1-Gracefulness on B(m,n,p) and B(m,n)

“除去4种特殊情况,连结两个顶点的3条独立路所成简单图B(m,n,p),是优美的”已被证明。本文提出k-优美图和k-GL矩阵的概念(k为非负整数),证明了这4种特殊情形,一种是优美的,其余是1-优美的。与此类似,设圈C_m=A_1A_2…A_mA_1,路P_n=A_1B_1B_2…B_n,本文还论述了C_m∪P_n的优美性。

We proved 'Apart from four exceptional cases, simple graphs B(m,n,p) consisting of three independent paths joining two vertices are graceful'. In this paper, we shall define that a simple graph G(V,E)is k-graceful (k is a nonnegative integer) if there is a labelling φ of its vertices with distinct integers from the set {0,1,2,…,|E|+k}, so that the induced edge labelling L defined by L(uv)=|φ(u)-φ(v)| assigns each edge a different label. We still give the concept of k-GL-matrix, by means of which we can prove that one of the four mentioned above is graceful and the other three are 1-graceful. And then we can prove that 'the graphs B(m,n)=C_m∪P_n are 1-graceful when m≡1 or 2(mod4); B(m,n) are graceful when m≡0 or 3(mod4), in which C_m=A_1A_2…A_mA_1, P_n=A_1B_1B_2…B_n(m≥3,n≥0).'