Towards generalizing MacDougall’s conjecture on vertex-magic total labelings

Abstract

MacDougall’s conjecture states that every regular graph of degree at least 2 has a vertex-magic total labeling (VMTL) with the lone exception of 2K3. Since there is enormous empirical evidence supporting this conjecture, it is reasonable to seek generalizations. Thus we ask the more general question: to what extent does the degree sequence of a graph determine the existence or nonexistence of a VMTL? We provide beginning steps towards answering this question, and related questions, by providing infinite families of degree sequences, and for each sequence, a graph with a VMTL and another graph without a VMTL.