Consecutive magic graphs

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+e1,2,…,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}{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}{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+n2=(2e+1)2(n-1)2+n2=(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-1a=n-1. Furthermore, we show that every a  -vertex consecutive magic graph has minimum degree at least two if a=0a=0, or both 2e⩾6n2-2n+1 and 2a⩽e2a⩽e, and the minimum degree is at least three if both 2e⩾10n2-6n+1+4a and 2a⩽e2a⩽e. Finally, we state analogous results for b-edge consecutive magic graphs.