Which Haar graphs are Cayley graphs?

Istv\'{a}n Est\'{e}lyi
University of Ljubljana, University of Primorska

Tomaž Pisanski
University of Primorska, University of Ljubljana

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Minisymposium: STRUCTURE AND PROPERTIES OF VERTEX-TRANSITIVE GRAPHS

Content: Haar graphs are regular covering graphs of dipoles. They can also be constructed in a Cayley graph-like manner, in terms of a group $G$ and its subset $S$. Just like Cayley graphs, these graphs frequently appear in various constructions, due to their pleasant symmetry properties. For example, Haar graphs were used for constructing families of semi-symmetric graphs and approximate cages. If $G$ is an abelian group, then its Haar graphs are well-known to be Cayley graphs; however, there are examples of non-abelian groups $G$ and subsets $S$ when this is not the case. Thus it might be worth investigating the Cayley-ness and more generally the vertex-transitivity of Haar graphs. In this talk I am going to address the problem of classifying finite non-abelian groups $G$ with the property that every Haar graph of $G$ is a Cayley graph. We will deduce an equivalent condition for a Haar graph of $G$ be a Cayley graph of a group containing $G$ in terms of $G$, $S$ and $\mathrm{Aut } G$. Moreover, the above problem will be solved for dihedral groups.

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