Graphs with prescribed local neighborhoods of their universal coverings
Charles Bordenave, Simon Coste

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.
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