On distance magic circulants of valency 6

Abstract

A graph $\Gamma =(V,E)$ of order $n$ is {\em distance magic} if it admits a bijective labeling $\ell :V\to \{1,2,\dots ,n\}$ of its vertices for which there exists a positive integer $\kappa$ such that ${\sum }_{u\in N(v)}\ell (u)=\kappa$ for all vertices $v\in V$, where $N(v)$ is the neighborhood of $v$. %It is well known that a regular distance magic graph is necessarily of even valency. A {\em circulant} is a graph admitting an automorphism cyclically permuting its vertices. In this paper we study distance magic circulants of valency $6$. We obtain some necessary and some sufficient conditions for a circulant of valency $6$ to be distance magic, thereby finding several infinite families of examples. The combined results of this paper provide a partial classification of all distance magic circulants of valency $6$. In particular, we classify distance magic circulants of valency $6$, whose order is not divisible by $12$.

19 pages

Cited by in corpus (2)

• Distance magic labelings of Cartesian products of cycles
• Self-reverse labelings of distance magic graphs