Checkerboard colourable twisted duals

TL;DR

This paper proves that every embedded graph has a checkerboard colourable twisted dual, and Eulerian embedded graphs have checkerboard colourable partial Petrials, using orientations of medial graphs and ribbon graph boundary components.

Contribution

It provides new proofs that every embedded graph has a checkerboard colourable twisted dual and Eulerian graphs have checkerboard colourable partial Petrials, answering prior open questions.

Findings

Every embedded graph has a checkerboard colourable twisted dual.

Eulerian embedded graphs have a checkerboard colourable partial Petrial.

Proofs utilize orientations of medial graphs and boundary components of ribbon graphs.

Abstract

In this note we show that any embedded graph has a checkerboard colourable twisted dual and any Eulerian embedded graph has a checkerboard colourable partial Petrial, answering questions posed by Ellis-Monaghan and Moffatt. The proofs are based on orientations of their medial graphs and orientations of boundary components of their corresponding ribbon graphs. The arrow presentations of ribbon graphs are also used. We also obtain two related results.