# Eulerian Spaces

## TL;DR

This paper unifies combinatorial and topological theories to define and characterize Eulerian spaces, extending known results to infinite graphs and continua, and solving a longstanding conjecture in the field.

## Contribution

It introduces a unified framework for Eulerian spaces, proves their equivalence across definitions, and advances the characterization of infinite Eulerian structures.

## Key findings

- Unified theory of Eulerian spaces established
- Conjecture for characterizing Eulerian spaces formulated
- Complete characterization of one-dimensional Eulerian spaces achieved

## Abstract

We develop a unified theory of Eulerian spaces by combining the combinatorial theory of infinite, locally finite Eulerian graphs as introduced by Diestel and K\"uhn with the topological theory of Eulerian continua defined as irreducible images of the circle, as proposed by Bula, Nikiel and Tymchatyn. First, we clarify the notion of an Eulerian space and establish that all competing definitions in the literature are in fact equivalent. Next, responding to an unsolved problem of Treybig and Ward from 1981, we formulate a combinatorial conjecture for characterising the Eulerian spaces, in a manner that naturally extends the characterisation for finite Eulerian graphs. Finally, we present far-reaching results in support of our conjecture which together subsume and extend all known results about the Eulerianity of infinite graphs and continua to date. In particular, we characterise all one-dimensional Eulerian spaces.