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On the degrees of a strongly vertex-magic graph

- Balbuena, Camino, Barker, Ewan, Das, K. C., Lin, Yuqing, Miller, Mirka, Ryan, Joe, Slamin,, Sugeng, Kiki Ariyanti, Tkac, M.

**Authors:**Balbuena, Camino , Barker, Ewan , Das, K. C. , Lin, Yuqing , Miller, Mirka , Ryan, Joe , Slamin, , Sugeng, Kiki Ariyanti , Tkac, M.**Date:**2006**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 306, no. 6 (2006), p. 539-551**Full Text:**false**Reviewed:****Description:**Let G=(V,E) be a finite graph, where |V|=n≥2 and |E|=e≥1. A vertex-magic total labeling is a bijection λ from V∪E to the set of consecutive integers {1,2,...,n+e} with the property that for every v∈V, λ(v)+∑w∈N(v)λ(vw)=h for some constant h. Such a labeling is strong if λ(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 2e≥10n2-6n+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. © 2006 Elsevier B.V. All rights reserved**Description:**C1**Description:**2003001603

- Balbuena, Camino, Barker, Ewan, Lin, Yuqing, Miller, Mirka, Sugeng, Kiki Ariyanti

**Authors:**Balbuena, Camino , Barker, Ewan , Lin, Yuqing , Miller, Mirka , Sugeng, Kiki Ariyanti**Date:**2006**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 306, no. 16 (2006), p. 1817-1829**Full Text:**false**Reviewed:****Description:**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)**Description:**C1**Description:**2003001604

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