On the degrees of a strongly vertex-magic graph Original Research Article

Let image be a finite graph, where image and image. A vertex-magic total labeling is a bijection image from image to the set of consecutive integers image with the property that for every image, image for some constant h. Such a labeling is strong if image. In this paper, we prove first that the minimum degree of a strongly vertex-magic graph is at least two. Next, we show that if image, then the minimum degree of a strongly vertex-magic graph is at least three. Further, we obtain upper and lower bounds of any vertex degree in terms of n and e. As a consequence we show that a strongly vertex-magic graph is maximally edge-connected and hamiltonian if the number of edges is large enough. Finally, we prove that semi-regular bipartite graphs are not strongly vertex-magic graphs, and we provide strongly vertex-magic total labeling of certain families of circulant graphs.