中文摘要：

设k为非负整数,G是一个p点q边图,如果将G的边用k,k＋1,k＋2,…,k＋q-1进行标号,而顶点标号模p运算后各不相同,则称G是k-边优美的.对于所有满足G为k-边优美图的非负整数k所构成的集合称为图G的边优美指标集.该文给出了图G=（V,E）为k-边优美的定义,根据轮图的特殊性质,讨论了S（3,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（3,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（3,n）的边优美指标集问题.

英文摘要：

Let kbe a nonnegative integer,and Gbe a graph with pvertices and qedges.The graph Gis called to be k-edge-graceful if there exists a bijection f：E→ {k,k＋1,k＋2,…,k＋q-1}such that the induced mapping f＋：V →Zpis a bijection too.We denote Gis k-edge-graceful.In this paper,the defination is given out which G=（V,E）is called to be k-edge-graceful graph.And by the especial property of the graph,the necessary condition which the graph S（3,n）is k-edge-graceful is discussed.A method to construct k-edge-graceful graph S（3,n）is given out by recursion and the problem of what sets of natural numbers are the edge-graceful indices of graph S（3,n）is completely resolved when nis even.