On distance magic circulants of valency 6

DOI10.1016/J.DAM.2022.12.024zbMATH Open1516.05196arXiv2203.09856MaRDI QIDQ6394053

Primož Šparl, Štefko Miklavič

Publication date: 18 March 2022

Abstract: A graph $Gamma=(V,E)$ of order $n$ is {em distance magic} if it admits a bijective labeling $ellcolonVo1,2,ldots,n$ of its vertices for which there exists a positive integer $kappa$ such that $su{m}_{uinN(v)}ell(u)=kappa$ for all vertices $vinV$, 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 $12$.

Mathematics Subject Classification ID

Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50) Graph labelling (graceful graphs, bandwidth, etc.) (05C78) Vertex degrees (05C07)

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