一个简单图G=(V,E)的反魔幻标号是一个双射f:E→{ 1,2,⋯,| E | },使得任意顶点所关联的边的标号之和互不相同。如果一个图存在魔幻标号,则称其为反魔幻图。Hartsfield和Ringel猜想除K2以外的所有树图都是反魔幻的。令T是一个非K2的树图,V2(T)是T中所有顶点度为2的顶点集合。Liang,Wong和Zhu证明了若由V2(T)所得的诱导子图是一条路径P,且T中所有不属于V2(T)里的顶点的度均为奇数,则T是反魔幻图。令vs是路径P的中间点,且v是不属于T的一个新的顶点。设T'是通过连接vs和v由T所构造的新树。本文证明了T'仍保持反魔幻性。Let G=(V,E)be a simple graph. A bijection f:E→{ 1,2,⋯,| E | }is called anti-magic if the sum of labels of the edges incident to any vertex is distinct. A graph is called anti-magic if there exists anti-magic labeling. Hartsfield and Ringel conjected that every tree other than K2has an anti-magic labeling. Let T be a tree not K2and V2(T)be the set of vertices of degree 2 in T. Liang, Wong and Zhu showed that if the induced subgraph of V2(T)is a path P and the degree of any vertex in V(T)\V2(T)is odd, then T is anti-magic. Suppose that vsis the middle vertex of P and v is a new vertex. T' is a new tree obtained from T by joining vsand v. In this paper, we prove that T' is also anti-magic.
