三路树P(2m,2m,2m)是边幻图的证明

A Proof of Three-path P(2m, 2m, 2m)Being Edge-magic Graph

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 ,3,… ,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) .文 [1 ]中猜测树是边幻图 .本文证明了三路树P(m ,n ,t)当m ,n ,t为偶数且相等时为边幻图 .

Let G be a graph with p vertices and q edges. Assume the vertices and edges of G are labelled by 1, 2, 3…, p+q such that each label is used exactly once. We define the valence vale of an edge to be the sum of the label of e plus the two labels pf the vertices incident with e. If a labelling of G is possible such that the valence vale for e is constant, we call the graph G is edge magic. Reference[1]proposed the conjecture: every tree is edge magic. In this paper, we prove that P(2m, 2m, 2m)is edge magic.