Vertex-Magic Total Labelings of Regular Graphs

In this paper it is shown that if a graph $G$ possesses a spanning subgraph $H$ with a strong vertex magic total labeling (VMTL) and $G-E(H)$ is even-regular, then $G$ also has a strong VMTL. Among other things, this is used to conclude that all Hamiltonian regular graphs of odd order possess strong VMTLs. A relationship is then demonstrated between regular graphs of even degree and sparse magic squares. We next consider cubic graphs of order $2n$ consisting of two 2-factors of order $n$, connected by a 1-factor (quasi-prisms). Based on McQuillan’s construction of VMTLs of such 3-regular graphs, VMTLs are derived for similar regular graphs of any odd degree. Finally, a construction is given for VMTLs of quartic graphs of order $4n+2$ consisting of two cycles of odd order $n$ connected by a 2-factor (simple quasi-anti-prisms), and based on this construction VMTLs are derived for similar regular graphs of any even degree.