On Regular Distance Magic Graphs of Odd Order

Abstract

Let G=(V,E) be a graph with n vertices. A bijection f:V→{1,2,…,n} is called a distance magic abeling f G if there exists an integer k such that ∑u∈N(v)f(u)=k for all v∈V, where N(v) is the set of all ertices 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 ronček, 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.