Graphs with prescribed local neighborhoods of their universal coverings

TL;DR

This paper establishes a precise condition under which a collection of rooted trees can represent the local neighborhoods of the universal covering of a graph, aiding in understanding graph coverings and local structure.

Contribution

It provides a necessary and sufficient criterion for collections of rooted trees to correspond to local neighborhoods in universal coverings of graphs.

Findings

Characterization of local neighborhoods in universal coverings

Necessary and sufficient conditions for collections of rooted trees

Insights into graph covering structures

Abstract

Given a collection of n rooted trees with depth h, we give a necessary and sufficient condition for this collection to be the collection of h-depth universal covering neighborhoods at each vertex.