舵轮图都是强协调图

一个有e条边的简单图G称为是强协调的,若有V(G)到{0,1,…,e-1}的单射h,使导出映射h~*:h~*(uv)=h(u)+h(v)是由E(G)到{1,2,…,e}的一个双射。舵轮图H_n是由含n个顶点的圈C_n内添加一个与C_n的每个顶点都相邻的顶点,且再在C_n的每个顶点上都添上一条悬挂边而得到的图。本文中证明了,所有舵轮图都是强协调图,因而回答了[2]中一个open问题。

Helm Graphs are Strong Harmonious Graphs

A simple graph G with e edges is said a strong harmonious graph if, there exists an injection h from V(G) to {0,1,...,e-1}, so that the induced function h~*: h~*(uv)=h(u)+h(v) is a bijeetion from E(G) to {1,2,...,e}. A helm graph H_n is obtained from C_n, the cycle with n vertices, by adding one extra vertex which is joining to every vertex of C_n, and adding a single pendent edge to each vertex of C_n. In this paper, it's proved that all helms are strong harmonious graphs, hence an open problem in [2] is answered.