S(7,n)的k-边优美的图标号

On the k-edge-graceful indices of S(7,n)

设图G=(V,E),其中|V|=p,|E|=q.对于k∈N,如果存在一个双射f:E→{k,k+1,…,k+q-1},使得它的导出映射f+:V→Zp,ua∑(u,v)∈Ef(u,v)mod p也是一个双射,则称图G是k-边优美的.对于所有的满足G为k-边优美图的非负整数k构成的集合称为图G的边优美指标集.本文根据轮图的特殊性质,讨论了S(7,n)为k-边优美图的必要条件.根据所得的必要条件,利用递归的方法构造S(7,n)的k-边优美图标号并给出详细证明,从而完全解决了当n为偶数时S(7,n)的边优美指标集问题.

Let k be an nonnegative integer,and G be a graph with p vertices and q edges. The graph G is said to be k-edge-graceful if there is a bijection f : E→{k,k +1 ,…, k + q -1}, such that the induced mapping f^+:V→Zp,u→(u,v)∈E∑f(u,v)mod p mod p is a bijection too.In this paper,by the especial property of the graph,the necessary condition which the graph S (7,n) is k-edge-graceful is discussed.A method to construct k-edge-graceful graph S (7,n) is given out by recursion and the problem of what sets of natural numbers are the edge-graceful indices of graph S (7,n) is completely resolved when n is even.