Equivalence of local balance conditions for Speiser graphs and planar bipartite maps

The discussion concerns three local balance conditions: the condition for Speiser graphs in Theorem, the condition in section \S, and the condition in section \S. Equivalence conjecture. The three local balance conditions are equivalent. This is suggested by the relationship between tessellation schemes, Speiser graphs, and Tomasini's characterization of finite branched self-covers of the sphere; the source does not state whether the equivalence has been proved.