Every even cycle of order at least 8 has a mirror labeling

Abstract

A mirror labeling of the cycle $C_n$ is a vertex-magic total labeling (VMTL) for the cycle $C_n$ with the property that if x is a vertex label, then $2n+1-x$ is an edge label, for each $1\le x\le 2n$. (Note that any mirror labeling for $C_n$ can be easily converted into an edge-magic total labeling for $C_n$ with the same property, and vice versa.) It has been known for decades that every odd cycle has a mirror labeling. Mirror labelings for $C_n$ with n even are considerably more difficult to construct generally, with only the case $n\equiv 2$ mod 8 having been provided. In this paper, we obtain mirror labelings for all remaining cases, namely $n\equiv 6$ mod 8, $n\ge 14$ and $n\equiv 0$ mod 4, $n\ge 8$.

This result has significant ramifications for the study of vertex-magic total labelings of graphs generally. A quarter century ago, James MacDougall provided his guiding conjecture positing that every regular graph of degree at least 2 has a VMTL, except for the disjoint union $2C_3$. Ian Gray showed that every Hamiltonian regular graph of odd order possesses a VMTL, and introduced mirror vertex-magic total labelings as a tool to obtain a similar, general result for even order regular graphs. However, a key technical part of his program was missing, namely, the existence of mirror VMTLs for even order cycles. A mirror labeling is a particular kind of mirror VMTL. Thus, the results of this work provide the missing piece required for Gray's program. It now follows, that any Hamiltonian $(4t+2)$-regular graph of any even order ($n\ge 8$, $t\ge 0$) must have a VMTL. This provides substantial new progress towards resolving MacDougall's Conjecture.