Consecutive magic graphs

Abstract

Let G be a graph of order n 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 = ( 2 e + 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 2 e ⩾ 6 n 2 - 2 n + 1 and 2 a ⩽ e , and the minimum degree is at least three if both 2 e ⩾ 10 n 2 - 6 n + 1 + 4 a and 2 a ⩽ e . Finally, we state analogous results for b-edge consecutive magic graphs.