A bijection for tri-cellular maps

In this paper we give a bijective proof for a relation between unicellular, bicellular
and tricellular maps. These maps represent cell-complexes of orientable surfaces having one, two or three boundary components.
The relation can formally be obtained using matrix theory \cite{Dyson} employing
the Schwinger-Dyson equation \cite{Schwinger}. In this paper we present a bijective proof of the corresponding coefficient equation. Our result is a bijection that transforms a unicellular map of genus $g$ into unicellular, bicellular or tricellular maps of strictly lower genera. The bijection employs edge-cutting, edge-contraction and edge-deletion.