三路树P(m,n,t)是边幻图的证明(II)

A Proof of Three-path Trees P(m,n,t) Being Edge-magic(II)

令简单图G=(V,E)是有p个顶点q条边的图.假设G的顶点和边由1,2,…,p+q所标号,且f:V∪E→{1,2,…,p+q}是一个双射,如果对所有的边xy,f(x)+f(y)+f(xy)是常量,则称图G是边幻图(edge-magic).本文证明了三路树P(m,n,t)当n为偶数,t=n+2时也是边幻图.

Let G be a graph with p vertices and q edges. Assume the vertices and edges of G are labeled by 1,2,…,(p+q) such that each label is used exactly once. We define the valence of an edge to be the sum of the label of e plus the two labels of the vertices incident with e. If a labeling of G is possible such that the valence for e is constant, we call the graph G is edge-magic. In this paper, we proof three-path tree P(m,n,t) is edge-magic when n is oven t=n+2.