Complete bipartite multi-graphs with a unique regular dessin

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.