Well begun is half done: the computation of Eulerian paths

Abstract

We characterize the uniform computational strength of finding one-way Eulerian paths in countable graphs. For both directed and undirected graphs, this problem is Weihrauch-equivalent to $\mathsf{lim}\times\mathsf{C}_{\mathbb{N}}'$, where $\mathsf{C}_{\mathbb{N}}'$ is the jump of closed choice on the natural numbers. This refines the proof-theoretical strength of the existence of an Eulerian path given by the reverse mathematics framework. The $\mathsf{C}_{\mathbb{N}}'$ component comes entirely from the difficulty of choosing a starting vertex: with a valid start prescribed, the problem is equivalent to $\mathsf{lim}$. We also study the problem $\mathsf{OrientEP}$ of orienting an undirected graph while preserving the existence of a one-way Eulerian path. We prove $\mathsf{lim}_2<_{\mathrm{W}}\mathsf{OrientEP}<_{\mathrm{W}}\mathsf{C}_{2}'\sqcap\mathsf{Sort}_2$ and $\mathsf{OrientEP}\not\le_{\mathrm{W}}\mathsf{C}_{\mathbb{N}}$.