# Three Graph Duals and A Bijection

## TL;DR

This paper introduces a unified notion of graph duals encompassing three different definitions, and uses this to reprove a bijection between vertex-labeled trees and permutation factorizations, extending previous work.

## Contribution

It generalizes the concept of graph duals beyond trees, providing three equivalent definitions and applying this to reprove a key combinatorial bijection.

## Key findings

- Unified three definitions of graph duals: graph-theoretic, algebraic, and combinatorial.
- Proved the equivalence of these duals and their relation to the topological dual.
- Reproved Goulden and Yong's bijection using the new dual concept.

## Abstract

We develop a notion of a dual of a graph, generalizing the definition of Goulden and Yong (which only applied to trees), and reproving their main result using our new notion. We in fact give three definitions of the dual: a graph-theoretic one, an algebraic one, and a combinatorial "mind-body" dual, showing that they are in fact the same, and are also the same (on trees) as the topological dual developed by Goulden and Yong. Goulden and Yong use their dual to define a bijection between the vertex labeled trees and the factorizations of the permutation $(n, \ldots, 1)$ into $n-1$ transpositions, showing that their bijection has a particular structural property. We reprove their result using our dual instead.