Orienting triangulations

TL;DR

This paper proves that for most surfaces, triangulations can be oriented to have no sinks and vertices with outdegree divisible by three, advancing the understanding of graph orientations on complex surfaces.

Contribution

It confirms a conjecture by Barát and Thomassen and extends Schnyder woods to higher genus surfaces.

Findings

Triangulations of surfaces (excluding sphere and projective plane) admit sink-free orientations with outdegree divisible by three.

The result supports generalizing Schnyder woods to higher genus surfaces.

Provides a constructive proof for the existence of such orientations.

Abstract

We prove that any triangulation of a surface different from the sphere and the projective plane admits an orientation without sinks such that every vertex has outdegree divisible by three. This confirms a conjecture of Bar\'at and Thomassen and is a step towards a generalization of Schnyder woods to higher genus surfaces.