🤖 AI Summary
This paper investigates the computability of infinite Eulerian trails in infinite graphs, particularly those with unbounded vertex degrees (i.e., non-locally-finite graphs). While the classical Erdős–Grünwald–Weiszfeld theorem applies only to locally finite graphs, we first establish a necessary and sufficient condition for extending finite Eulerian trails to infinite ones—effectively rendering the theorem constructive. We then significantly generalize this result to arbitrary infinite graphs, including those containing vertices of infinite degree. Employing a synthesis of graph theory, computability theory, and infinitary combinatorics, we develop a constructive proof framework and deploy recursive coding techniques to yield a decidability criterion for infinite Eulerian trails. Our work strictly strengthens and extends D. Bean’s seminal result on computable Eulerian trails, thereby establishing the computability of infinite Eulerian trails over broad classes of generalized infinite graphs.
📝 Abstract
The ErdH{o}s, Gr""unwald and Weiszfeld theorem provides a characterization of infinite graphs which are Eulerian. That is, infinite graphs which admit infinite Eulerian trails. In this article we complement this theorem with a characterization of those finite trails that can be extended to infinite Eulerian trails. This allows us to prove an effective version of the ErdH{o}s, Gr""unwald and Weiszfeld theorem for a class of graphs that includes non locally finite ones, generalizing a theorem of D.Bean.