A regular dessin is an orientable edge-regular bipartite map. Jones et al.(J Combin Theory Ser B 98:241–248, 2008) showed that a complete bipartite graph \(\mathbf{K}_{n,n}\) has a unique orientably (arc-)regular map if and only if \(\gcd (n,\phi (n))=1\). We extended this result in Fan and Li (J Graph Theory 87:581–586, 2018) by proving that a complete bipartite graph \(\mathbf{K}_{m,n}\) underlies a unique regular dessin if and only if \(\gcd (m,\phi (n))=1\) and \(\gcd (n,\phi (m))=1\). In this paper, it is shown that a complete bipartite multi-graph \(\mathbf{K}_{m,n}^{(\lambda )}\) with \(\lambda >1\) underlies a unique regular dessin if and only if \(\lambda =2\), \(\gcd (m_2,n_2)=1\), and \(\gcd (m,\phi (n_{2'}))\gcd (n,\phi (m_{2'}))=1\), where \(n_{2}\) and \(n_{2'}\) denote the 2-part and \(2'\)-part of n, respectively. Furthermore, apart from two degenerated cases, each of such dessins can be uniquely decomposed into the direct product of two dessins such that one is symmetric and reflexible, and the other has only one face.

规则的dessin是可定向的边缘-规则的二分图。Jones等人（J Combin Theory Ser B 98：241–248，2008）表明，如果一个完整的二部图\（\ mathbf {K} _ {n，n} \）具有唯一的定向（弧形）正则图，并且仅当\（\ gcd（n，\ phi（n））= 1 \）时。我们通过证明一个完整的二部图\（\ mathbf {K} _ {m，n} \）在唯一且唯一的常规设计基础上在Fan and Li（J Graph Theory 87：581–586，2018）中扩展了这一结果如果\（\ gcd（m，\ phi（n））= 1 \）和\（\ gcd（n，\ phi（m））= 1 \）。本文显示了一个完整的二部多重图\（\ mathbf {K} _ {m，n} ^ {（\ lambda}} \）和\（\ lambda> 1 \）是唯一的规则设计当且仅当\（\ lambda = 2 \），\（\ gcd（m_2，n_2）= 1 \）和\（\ gcd（m，\ phi（n_ {2'}）））\ gcd（n，\ phi（m_ {2 '}））= 1 \） ，其中\（N_ {2} \）和\（N_ {2'} \）表示2个部分和\（2' \）的双组分ñ分别。此外，除了两个退化的情况之外，每个这样的dessins可以被独特地分解为两个dessins的直接产物，使得一个是对称且可弯曲的，而另一个仅具有一个面。