Circular regular dessins with underlying graphs ${\bf K}_{2^e,2^f}^{(2^d)}$: the metacyclic case
A dessin is a 2-cell embedding of a connected bipartite graph into an orientable closed surface. It is regular if the colour- and orientation-preserving automorphism group acts regularly on edges, and circular if every face boundary is a simple cycle. We classify and enumerate circular regular dessins of the complete bipartite multigraph ${\bf K}_{2^e,2^f}^{(2^d)}$ ($e\ge f\ge 2$, $d\ge 1$) whose automorphism group is {\it metacyclic}. This extends the work of Fan (circular, metacyclic) and Cai--Hu (regular, metacyclic) from simple graphs ${\bf K}_{2^e,2^f}$ to multigraphs ${\bf K}_{2^e,2^f}^{(2^d)}$. Such dessins correspond to non-exact bicyclic factorisations $G=ła\rłb\r\cong\ZZ_{2^{e+d}}\ZZ_{2^{f+d}}$, where $ła\r\capłb\r\cong\ZZ_{2^d}$ and $łba\r\cap ła\r=łba\r\cap łb\r=1$. In general, a group admitting such a factorisation need not be metacyclic; in this paper we impose the additional condition that $G$ is a metacyclic 2-group. Under this assumption, we obtain explicit enumeration formulas for the number of such dessins up to isomorphism. Together with the known classifications for simple graphs (the case $d=0$), this completes the enumeration of the metacyclic case for 2-power complete bipartite multigraphs.