Congruence distance magic graphs

In this paper, we introduce a series of necessary conditions namely the concept of $p$-distance magic labeling and give a relation between labeling and graph-theoretic properties of a graph. We also prove that a graph which is $p$-distance magic, for all $p$ is also distance magic. As a result we have a new necessary and sufficient condition for a distance magic graph. We also obtain conditions for when the ``group distance magic constant will be unique".