# Triangulations with homogeneous zigzags

## TL;DR

This paper explores the relationship between triangulations with homogeneous zigzags and directed Eulerian graphs, introducing a new family of spherical triangulations with a single zigzag and homogeneous properties.

## Contribution

It establishes a one-to-one correspondence between homogeneous zigzag triangulations and Eulerian graph embeddings, and constructs a new family of z-knotted spherical triangulations.

## Key findings

- Established correspondence between triangulations with homogeneous zigzags and Eulerian graph embeddings.
- Constructed a family of tree-structured z-knotted spherical triangulations.
- Showed that certain triangulations have a single zigzag (z-knotted).

## Abstract

We investigate zigzags in triangulations of connected closed $2$-dimensional surfaces and show that there is a one-to-one correspondence between triangulations with homogeneous zigzags and closed $2$-cell embeddings of directed Eulerian graphs in surfaces. A triangulation is called $z$-knotted if it has a single zigzag. We construct a family of tree structured $z$-knotted spherical triangulations whose zigzags are homogeneous.