Tetravalent distance magic graphs of small order and an infinite family of examples

A graph of order n is distance magic if it admits a bijective labeling of its vertices with integers from 1 to n such that each vertex has the same sum of the labels of its neighbors.This paper contributes to the long term project of characterizing all tetravalent distance magic graphs. With the help of a computer we find that out of almost nine million connected tetravalent graphs up to order 16 only nine are distance magic. In fact, besides the six well known wreath graphs there are only three other examples, one of each of the orders 12, 14 and 16. We introduce a generalization of wreath graphs, the so-called quasi wreath graphs, and classify all distance magic graphs among them. This way we obtain infinitely many new tetravalent distance magic graphs. Moreover, the two non-wreath graphs of orders 12 and 14 are quasi wreath graphs while the one of order 16 can be obtained from a quasi wreath graph of order 14 using a simple construction due to Kovář, Fronček and Kovářová.