On the degrees of a strongly vertex-magic graph

Abstract

Let G=(V ,E) be a finite graph, where |V |=n2 and |E|=e1.A vertex-magic total labeling is a bijection from V ∪E to the set of consecutive integers {1, 2, . . . , n + e} with the property that for every v ∈ V , (v) +w∈N(v) (vw) = h for some constant h. Such a labeling is strong if (V )={1, 2, . . . , n}. 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 2e10n2 − 6n + 1, 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.