In this paper we are concerned with the classification of the finite groups admitting a bipartite DRR and a bipartite GRR. First, we find a natural obstruction in a finite group for not admitting a bipartite GRR. Then we give a complete classification of the finite groups satisfying this natural obstruction and hence not admitting a bipartite GRR. Based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite GRR. Next, we prove the existence of bipartite DRRs for most of the finite groups not admitting a bipartite GRR found in this paper. Actually, we prove a much stronger result: we give an asymptotic enumeration of the bipartite DRRs over these groups. Again, based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite DRR.

中文翻译：

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关于二部图形正则表示的猜想

在本文中，我们关注允许二分 DRR 和二分 GRR 的有限群的分类。首先，我们在有限群中发现了一个自然障碍，因为它不接受二分 GRR。然后我们给出满足这种自然障碍的有限群的完整分类，因此不接受二部 GRR。基于这些结果和一些广泛的计算机计算，我们陈述了一个猜想，旨在给出允许二分 GRR 的有限群的完整分类。接下来，我们证明了大多数不承认本文中发现的二分 GRR 的有限群的二分 DRR 的存在。实际上，我们证明了一个更强的结果：我们对这些组的二分 DRR 进行了渐近枚举。同样，基于这些结果和一些广泛的计算机计算，