Framed 4-Valent Graphs: Euler Tours, Gauss Circuits and Rotating Circuits

TL;DR

This paper provides an explicit formula to identify and describe Gauss circuits on framed 4-valent graphs based on Euler tours, with implications for understanding their existence and structure.

Contribution

It introduces a formula linking Euler tours and Gauss circuits on framed 4-valent graphs, enabling direct analysis from adjacency matrices.

Findings

The formula determines the existence of Gauss tours.

It applies to all symmetric matrices, not just chord diagram realizations.

Provides a method to describe unique Gauss circuits from Euler tours.

Abstract

In the present paper we give an explicit formula which allows us immediately to describe a unique Gauss circuit on a framed 4-valent graph (a graph with a structure of opposite edges) from an arbitrary Euler tour on the graph whenever the Gauss circuit exists. This formula only depends on the adjacency matrix of an Euler tour and also tells us whether there exists a Gauss tour on a framed 4-valent graph or not. It turns out that the results are also valid for all symmetric matrices (not just realisable by a chord diagram).