Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence of non-Cayley Haar graphs   A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that its
automorphism group ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$
acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple
bipartite graph $\Sigma$ such that ${\rm Aut}(\Sigma)$ contains a subgroup
isomorphic to $H$ acting semiregularly on $V(\Sigma)$ and the $H$-orbits are
equal to the partite sets of $\Sigma$. It is well-known that every Haar graph
of finite abelian groups is a Cayley graph. In this paper, we prove that every
finite non-abelian group admits a non-Cayley Haar graph except the dihedral
groups $D_6$, $D_8$, $D_{10}$, the quaternion group $Q_8$ and the group
$Q_8\times\mathbb{Z}_2$. This answers an open problem proposed by Est\'elyi and
Pisanski in 2016.

8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess