A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that its automorphism group $\text{Aut}\left(\Gamma \right)$ contains a subgroup isomorphic to $H$ acting regularly on $V\left(\Gamma \right)$, while a Haar graph of $H$ is a finite simple bipartite graph $\Sigma$ such that $\text{Aut}\left(\Sigma \right)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V\left(\Sigma \right)$ 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 Estélyi and Pisanski in 2016.

一组的Cayley图 $H$ 是一个有限的简单图 $\Gamma$ 这样它的自同构群 $\text{ut}（\Gamma ）$ 包含一个同构的亚组 $H$ 定期行动 $V（\Gamma ）$，而 $H$ 是有限的简单二部图 $\Sigma$ 这样 $\text{ut}（\Sigma ）$ 包含一个同构的亚组 $H$ 半定期地作用于 $V（\Sigma ）$ 和 $H$-轨道等于 $\Sigma$。众所周知，每个有限阿贝尔群的Haar图都是Cayley图。在本文中，我们证明除二面体组外，每个有限的非阿贝尔群都承认一个非凯利Haar图$d_6$， $d_8$， $d_{10}$四元数组 ${问}_8$ 和小组 ${问}_8\times {\mathbb{ž}}_2$。这回答了Estélyi和Pisanski在2016年提出的一个开放性问题。