A bijection for tri-cellular maps

TL;DR

This paper presents a bijective proof connecting uni-, bi-, and tricellular maps of specific topological genus, providing a combinatorial perspective on a relation typically derived from matrix theory.

Contribution

It introduces a novel bijection that explicitly constructs the relation between different types of maps, complementing matrix-theoretic approaches.

Findings

Established a bijective correspondence between map types

Provided a combinatorial proof of a topological relation

Enhanced understanding of map structures through explicit construction

Abstract

In this paper we give a bijective proof for a relation between uni- bi- and tricellular maps of certain topological genus. While this relation can formally be obtained using Matrix-theory as a result of the Schwinger-Dyson equation, we here present a bijection for the corresponding coefficient equation. Our construction is facilitated by repeated application of a certain cutting, the contraction of edges, incident to two vertices and the deletion of certain edges.