# On the CI-property for balanced configurations

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Sergio Hiroki Koike Quintanar

University of Primorska

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Istv\'an Kov\'acs

University of Primorska

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Dragan Marušič

University of Primorska

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Misha Muzychuk

Netanya Academic College

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**Minisymposium:**
APPLICATIONS OF GROUPS IN GRAPH THEORY

**Content:**
Let $G$ be a finite group. A Cayley object is a relational structure with underlying set $G$ which is invariant under all right translations $x \mapsto xg$, $x,g \in G$.
Given a class $\mathcal{K}$ of Cayley objects of $G$, the group $G$ is said to satisfy the CI-property for $\mathcal{K}$ if for any two isomorphic Cayley objects in $\mathcal{K}$, there is an isomorphism between them, which is at the same time an automorphism of the group $G$.
The question whether a given group $G$ satisfies the CI-property for a given class $\mathcal{K}$ of Cayley objects has been studied under various choices of $G$ and $\mathcal{K}$.
In this talk, we focus our attention to the class of balanced configurations where we show that every cyclic group satisfy the CI-property.