In this paper, we are studying vertex-magic total labelings (VMTLs) of simple graphs. By now much is known about methods for constructing VMTLs for regular graphs. Here we are studying non-regular graphs. We show how to construct labelings for several families of non-regular graphs, including graphs formed as the disjoint union of two other graphs already possessing VMTLs. We focus on conditions which make these VMTLs strong, so that previously known methods can then be used to build larger graphs from these which will themselves have VMTLs. In the second part of the paper, we investigate ways of describing how far a graph may be from being regular but still possess a VMTL.
Australasian Journal of Combinatorics Vol. 46, p. 173-183