Consecutive magic graphs

Let G be a graph of order it and size e. A vertex-magic total labeling is an assignment of the integers 1, 2,..., n + e to the vertices and the edges of G, so that at each vertex, the vertex label and the labels on the edges incident at that vertex, add to a fixed constant, called the magic number of G. Such a labeling is a-vertex consecutive magic if the set of the labels of the vertices is {a + 1, a + 2,..., a + n}, and is b-edge consecutive magic if the set of labels of the edges is {b + 1, b + 2,..., b + e}. In this paper we prove that if an a-vertex consecutive magic graph has isolated vertices then the order and the size satisfy (n - 1)(2) + n(2) = (2e + 1)(2). Moreover. we show that every tree with even order is not a-vertex consecutive magic and, if a tree of odd order is a-vertex consecutive then a = n - 1. Furthermore, we show that every a-vertex consecutive magic graph has minimum degree at least two if a = 0, or both 2e >= root 6n(2) - 2n + 1 and 2a <= e, and the minimum degree is at least three if both 2e <= root 10n(2) - 6n + 1 + 4, and 2a <= e. Finally, we state analogous results for b-edge consecutive magic graphs. (c) 2006 Elsevier B.V. All rights reserved.