A Technique for Constructing Magic Labelings of 2-Regular Graphs

Abstract

A new technique is given for constructing a vertex-magic total labeling, and hence an edge-magic total labeling, for certain finite simple $2$-regular graphs. Let $C_r$ denote the cycle of length $r$. Let $n$ be an odd positive integer with $n=2m+1$. Let $k_i$ denote an integer such that $k_i\ge 3$, for $i=1,2,\dots ,l$, and write $n{C}_{k_i}$ to mean the disjoint union of $n$ copies of $C_{k_i}$. Let $G$ be the disjoint union $G\cong C_{k_1}\cup \dots \cup C_{k_l}$. Let $I=\{1,2,\dots ,l\}$ and let $J$ be any subset of $I$. Finally, let $G_J=\left(\bigcup _{i\in J}n{C}_{k_i}\right)\cup \left(\bigcup _{i\in I–J}{C}_{n{k}_i}\right)$, where all unions are disjoint unions. It is shown that if $G$ has a vertex-magic total labeling (VMTL) with a magic constant of $h$, then $G_J$ has VMTLs with magic constants $6m(k_1+k_2+\dots +k_l)+h$ and $nh–3m$. In particular, if $G$ has a strong VMTL then $G_J$ also has a strong VMTL.