Infinite Eulerian paths are computable on graphs with vertices of infinite degree
Nicanor Carrasco-Vargas
TL;DR
The paper extends the Erdős–Grünwald–Weiszfeld theorem to graphs that may have vertices of infinite degree by introducing moderately computable graphs, in which the vertex-degree function is computable. It provides a constructive characterization of finite paths that can be extended to infinite Eulerian trails and proves that, under the EGW hypotheses, there exist computable one-way or two-way infinite Eulerian paths; moreover, the problem of extending a given finite path is decidable. The results yield uniform algorithms for the two-way case and a slightly nonuniform but effective construction for the one-way case, offering the first positive computability result for graphs with infinite degree. This contributes to Nerode-style computability programs in graph theory and strengthens the bridge between classical infinite Eulerian theory and algorithmic realizability.
Abstract
The Erdős, Grünwald, and Weiszfeld theorem is a characterization of those infinite graphs which are Eulerian. That is, infinite graphs that admit infinite Eulerian paths. In this article we prove an effective version of the Erdős, Grünwald, and Weiszfeld theorem for a class of graphs where vertices of infinite degree are allowed, generalizing a theorem of D.Bean. Our results are obtained from a characterization of those finite paths in a graph that can be extended to infinite Eulerian paths.