On irregular total labellings
- Authors: Baca, Martin , Jendrol, Stanislav , Miller, Mirka , Ryan, Joe
- Date: 2007
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 307, no. 11-12 (May 2007), p. 1378-1388
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- Description: Two new graph characteristics, the total vertex irregularity strength and the total edge irregularity strength, are introduced. Estimations on these parameters are obtained. For some families of graphs the precise values of these parameters are proved. (c) 2006 Elsevier B.V. All rights reserved.
- Description: C1
- Description: 2003004909
On the degrees of a strongly vertex-magic graph
- 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
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- 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