TY - JOUR
T1 - On the degrees of a strongly vertex-magic graph
AU - Balbuena, C.
AU - Barker, E.
AU - Das, K. C.
AU - Lin, Y.
AU - Miller, M.
AU - Ryan, J.
AU - Slamin,
AU - Sugeng, K.
AU - Tkac, M.
PY - 2006/4/6
Y1 - 2006/4/6
N2 - 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.
AB - 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.
KW - Degree
KW - Graph
KW - Labeling
KW - Supervertex-magic
UR - http://www.scopus.com/inward/record.url?scp=33645402535&partnerID=8YFLogxK
U2 - 10.1016/j.disc.2006.01.019
DO - 10.1016/j.disc.2006.01.019
M3 - Article
AN - SCOPUS:33645402535
VL - 306
SP - 539
EP - 551
JO - Discrete Mathematics
JF - Discrete Mathematics
SN - 0012-365X
IS - 6
ER -