Vertex-magic labeling of trees and forests

A vertex-magic total labeling of a graph G(V; E) is a one-to-one map from E ∪V onto the integers {1; 2; : : : ; |E|+ |V |} such that (x) + ∑ (xy); where the sum is over all vertices y adjacent to x, is a constant, independent of the choice of vertex x. In this paper we examine the existence of vertex-magic total labelings of trees and forests. The situation is quite di9erent from the conjectured behavior of edge-magic total labelings of these graphs. We pay special attention to the case of so-called galaxies, forests in which every component tree is a star. c © 2002 Elsevier Science B.V. All rights reserved. 0. Introduction All graphs in this paper will be 4nite. The graph G=G(V; E) has vertex-set V =V (G) and edge-set E=E(G); we write v for |V (G)| and e for |E(G)|. A total labeling is a one-to-one map from E∪V onto the integers {1; 2; : : : ; e+v}. The weight of vertex x is the value (x)+ ∑ (xy) (where the sum is over all vertices y adjacent to x), and the weight of edge xy is (x) + (xy) + (y). A total labeling is edge-magic if there is a constant k such that every edge xy has weight k, and vertex∗ Corresponding author. Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA. E-mail address: [email protected] (W.D. Wallis). 0012-365X/03/$ see front matter c © 2002 Elsevier Science B.V. All rights reserved. PII: S 0012 -365X(02)00475 -2 286 I.D. Gray et al. / Discrete Mathematics 261 (2003) 285–298 magic if there is a constant h such that every vertex x has weight h. A graph with an edge-magic total labeling is called edge-magic, and k is called the magic sum associated with ; similarly a graph with a vertex-magic total labeling is vertex-magic, and h is the magic constant. Kotzig and Rosa [2] introduced edge-magic total labelings, under the name “magic valuations”. In particular, they showed that all caterpillars are edge-magic, and conjectured that all trees are edge-magic. This conjecture is interesting because of its similarity to the long-standing conjecture that all trees have graceful labelings, but so far there has been no progress on it. Vertex-magic total labelings were de4ned in [3], after MacDougall observed that this natural analog of the edge-magic case arose in the solution to a high-school enrichment problem [4]. We shall see that not all trees are vertex-magic, and also explore results about forests. 1. Trees In discussing trees, it is common to de4ne a leaf to be a vertex of degree 1. Other vertices are called internal. The vertex-magic property depends on the proportion of leaves. Theorem 1. Let T be a tree with n internal vertices and n leaves. Then T does not admit a vertex-magic total labeling if ¿ 1 + √ 12n2 + 4n+ 1 2n : Proof. If T has n internal vertices and n leaves, then v=( +1)n and e=( +1)n−1. So the labels to be used are {1; 2; : : : ; M} where M=2( + 1)n− 1. The maximum possible sum of weights on the leaves will be the sum of the 2 n largest labels: