James A. MacDougall, I. D. Gray

ABSTRACT

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 $r \geq 4$, every $r$-regular graph of odd order $n \leq 17$ 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.