The Locating-Chromatic number of an $n$-ary Trees

TL;DR

This paper investigates the asymptotic behavior of the locating-chromatic number in $n$-ary trees, revealing different growth patterns depending on whether the tree's height or branching factor tends to infinity.

Contribution

It provides a detailed analysis of how the locating-chromatic number varies with the parameters of $n$-ary trees, highlighting distinct asymptotic behaviors.

Findings

For fixed height, the locating-chromatic number approaches $n + k - 1$ as $n$ increases.

When the branching factor is fixed, the locating-chromatic number grows slower than $k$, specifically $o(k)$.

The behavior of the locating-chromatic number differs significantly between increasing height and increasing branching factor.

Abstract

The locating-chromatic number of a graph $G$ is the smallest integer $n$, such that $G$ has a proper $n$-coloring $c$ and all vertices have different vectors of distances to the colors generated by $c$. We study the asymptotic value of the locating-chromatic number of a $k$-level $n$-ary tree. The locating-chromatic number of this tree acts very differently when $k$ goes to infinity and when $n$ goes to infinity. If we fix $k\geq2$, almost all $n$-ary Tree $T(n,k)$ satisfy $\chi_L(T(n,k))=n+k-1$; so $\lim\limits_{n\to \infty} \chi_L(T(n,k))-n=k-1$. But if we fix $n\geq 2$, then $\chi_L(T(n,k))=o(k)$.