Tereza Škublová: On distance magic circulants of higher valencies

On distance magic circulants of higher valencies Tereza Škublová

Abstract: A circulant $\operatorname{Circ}(n,S)$ is a Cayley graph $\operatorname{Cay}(G,S)$, where $G$ is the finite cyclic group $\mathbb{Z}_n$. We will suppose that $S \neq \varnothing$, $S = -S$ and $\langle S \rangle = \mathbb{Z}_n$. We call a circulant $G = \operatorname{Circ}(n,S)$ distance magic if there exists a bijection $\ell$ from the vertex set of $G$ to the set ${1,2,\ldots,n}$ such that for each vertex $x$ the sum of values of function $\ell$ through the vertices adjacent to vertex $x$ is constant for all vertices $x$ of $G$.

Štefko Miklavič and Primož Šparl gave us a partial classification of the $6$-regular distance magic circulants in their article On distance magic circulants of valency $6$, Discrete Applied Mathematics, 2022. In this talk we will give a brief overview of their results and discuss generalizations of their theorems to $2p$-regular circulants for an odd prime $p$.