On Regular Distance Magic Graphs of Odd Order

Abstract

Let $G=(V,E)$ be a graph with $n$ vertices. A bijection $f:V\to \{1,2,\dots ,n\}$ is called a distance magic labeling of $G$ if there exists an integer $k$ such that $\sum _{u\in N(v)}f(u)=k$ for all $v\in V$, where $N(v)$ is the set of all vertices adjacent to $v$. Any graph which admits a distance magic labeling is a distance magic graph. The existence of regular distance magic graphs of even order was solved completely in a paper by Fronček, Kovář, and Kovářová. In two recent papers, the existence of $4$-regular and of $(n-3)$-regular distance magic graphs of odd order was also settled completely. In this paper, we provide a similar classification of all feasible odd orders of $r$-regular distance magic graphs when $r=6,8,10,12$. Even though some nonexistence proofs for small orders are done by brute force enumeration, all the existence proofs are constructive.