# Characterizations of Eulerian and even-face partial duals of ribbon graphs

## TL;DR

This paper extends characterizations of Eulerian and even-face partial duals from plane graphs to all orientable ribbon graphs, simplifying the criteria using crossing-total directions of medial graphs.

## Contribution

It generalizes existing results from plane graphs to orientable ribbon graphs and introduces simpler crossing-total directions for characterizing Eulerian partial duals.

## Key findings

- Extension of Huggett and Moffatt's result to orientable ribbon graphs
- Counterexample showing non-extension to non-orientable graphs
- Simplified characterization of Eulerian partial duals using crossing-total directions

## Abstract

Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of all-crossing directions of its medial graph. Then Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. Plane graphs are ribbon graphs with genus 0. In this paper, we shall first extend Huggett and Moffatt's result to any orientable ribbon graph and provide an example to show that it is not true for non-orientable ribbon graphs. Then we characterize all Eulerian partial duals of any ribbon graph in terms of crossing-total directions of its medial graph, which are much more simple than semi-crossing directions.