On regular distance magic graphs of odd order (Q6580123)

Let \(G\) be a graph on \(n\) vertices. A bijection \(f: V(G) \rightarrow \{1,\ldots, n\}\) 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 \(v\in V(G)\). A graph is distance magic if it has a distance magic labeling. The existence of regular distance magic graphs of even order was solved completely in [\textit{D. Fronček} et al., Bull. Inst. Comb. Appl. 48, 31--33 (2006; Zbl 1103.05037)]. When \(n\) is odd, the existence of \(r\)-regular distance magic graphs is not completely solved. The case when \(r=4\) was done in [\textit{P. Kovář} et al., Australas. J. Comb. 54, 127--132 (2012; Zbl 1278.05214)] and the case when \(r=n-3\) was done in [\textit{P. Kovář} and \textit{A. Silber}, AKCE Int. J. Graphs Comb. 9, No. 2, 213--219 (2012; Zbl 1256.05193)]. The authors answer the question for \(r\in \{6, 8, 10, 12\}\).