Self-reverse distance magic labelings

According to a nonstandard definition introduced in 2021, a distance magic labelling ℓ of a regular graph of order n is a bijection from its vertex set to the set of integers of the arithmetic progression from 1-n to n-1 with common difference 2, such that the sum of the labels of the neighbors of each vertex is zero. Such a labeling is called self-reverse if, for any pair of vertices u and v, u is adjacent to v if and only if the vertices with labels -ℓ(u) and -ℓ(v) are adjacent.

In this talk, we present the motivation for studying self-reverse distance magic labelings. We focus on self-reverse distance magic labelings in the case of tetravalent graphs. First, we examine some tetravalent distance magic graphs of small order and determine  which of them admit a self-reverse distance magic labeling. We then provide several examples and a complete classification of all orders for which a tetravalent graph admitting such a labeling exists. The classification is obtained via a novel construction that produces a (tetravalent) distance magic graph from two given (tetravalent) distance magic graphs. We also discuss the existence of graphs admitting a self-reverse distance magic labeling among some well-known families of tetravalent graphs and conclude with possible directions for future research.

This is joint work with Petr Kovář and Primož Šparl.