On distance magic circulants of valency 6

TL;DR

This paper investigates the conditions under which circulant graphs of valency 6 are distance magic, providing partial classifications and identifying infinite families of such graphs.

Contribution

It offers necessary and sufficient conditions for distance magic circulants of valency 6, advancing the classification of these graphs.

Findings

Identified infinite families of distance magic circulants of valency 6.

Provided partial classification for circulants whose order is not divisible by 12.

Established conditions distinguishing 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 $\ell \colon V \to \{1,2, \ldots, n\}$ 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 $12$.