Asymptotic rays

TL;DR

This paper characterizes graphs that are asymptotically similar to a ray, showing they must be uniformly spherically bounded with bounded local degrees, resolving a problem in combinatorics.

Contribution

It provides a necessary and sufficient condition for a graph to be asymptotically isomorphic to a ray, addressing a longstanding open problem.

Findings

Graphs asymptotically isomorphic to a ray are exactly those with uniform spherical bounds and bounded local degrees.

The paper solves Problem 10.1 from [3] in combinatorics.

It establishes a clear structural characterization of such graphs.

Abstract

We prove that a graph G is asymptotically isomorphic to the ray if and only if G is uniformly spherically bounded and is of bounded local degrees. This problem arouse in combinatorics and was posed in [3] (Problem 10.1).