Vertex-magic labelings of regular graphs II

Previously the first author has shown how to construct vertex-magic total labelings (VMTLs) for large families of regular graphs. The construction proceeds by successively adding arbitrary 2-factors to a regular graph of order n which possesses a strong VMTL, to produce a regular graph of the same o...

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Bibliographic Details
Published in:Discrete mathematics 2009-10, Vol.309 (20), p.5986-5999
Main Authors: Gray, I.D., MacDougall, J.A.
Format: Article
Language:English
Subjects:
Combinatorics
Combinatorics. Ordered structures
Exact sciences and technology
Graph theory
Magic labeling
Mathematics
Sciences and techniques of general use
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Summary:Previously the first author has shown how to construct vertex-magic total labelings (VMTLs) for large families of regular graphs. The construction proceeds by successively adding arbitrary 2-factors to a regular graph of order n which possesses a strong VMTL, to produce a regular graph of the same order but larger size. In this paper, we exploit this construction method. We are able to show that for any r ≥ 4 , every r -regular graph of odd order n ≤ 17 has a strong VMTL. We show how to produce strong labelings for some families of 2-regular graphs since these are used as the starting points of our construction. While even-order regular graphs are much harder to deal with, we introduce ‘mirror’ labelings which provide a suitable starting point from which the construction can proceed. We are able to show that several large classes of r -regular graphs of even order (including some Hamiltonian graphs) have VMTLs.
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2009.04.031