On the locating chromatic number of infinite trees

TL;DR

This paper investigates the conditions under which infinite trees have a finite locating chromatic number and introduces an algorithm for computing such colorings applicable to both finite and infinite trees.

Contribution

It provides necessary and sufficient conditions for infinite trees to have finite locating chromatic numbers and presents a universal algorithm for locating colorings.

Findings

Identified conditions for finite locating chromatic numbers in infinite trees.

Developed an algorithm for locating coloring applicable to finite and infinite trees.

Enhanced understanding of vertex identification in infinite graph structures.

Abstract

The locating chromatic number of a graph is the smallest integer $n$ such that there is a proper $n$-coloring $c$ and every vertex has a unique vector of distances to colors in $c$. We explore the necessary conditions and provide sufficient conditions for an infinite tree to have a finite locating chromatic number. We also give an algorithm for computing the locating coloring of trees that works for both finite and infinite trees.