Characterizing All Trees with Locating-chromatic Number 3

Let $c$ be a proper $k$-coloring of a connected graph $G$. Let $\Pi = \{S_{1}, S_{2},\ldots, S_{k}\}$ be the induced partition of $V(G)$ by $c$, where $S_{i}$ is the partition class having all vertices with color $i$.The color code $c_{\Pi}(v)$ of vertex $v$ is the ordered$k$-tuple $(d(v,S_{1}), d(v,S_{2}),\ldots, d(v,S_{k}))$, where$d(v,S_{i})= \hbox{min}\{d(v,x)|x \in S_{i}\}$, for $1\leq i\leq k$.If all vertices of $G$ have distinct color codes, then $c$ iscalled a locating-coloring of $G$.The locating-chromatic number of $G$, denoted by $\chi_{L}(G)$, isthe smallest $k$ such that $G$ posses a locating $k$-coloring. Clearly, any graph of order $n \geq 2$ have locating-chromatic number $k$, where $2 \leq k \leq n$. Characterizing all graphswith a certain locating-chromatic number is a difficult problem. Up to now, we have known allgraphs of order $n$ with locating chromatic number $2, n-1,$ or $n$.In this paper, we characterize all trees whose locating-chromatic number $3$. We also give a family of trees with locating-chromatic number 4

Trees with Certain Locating-chromatic Number

The locating-chromatic number of a graph G can be defined as the cardinality of a minimum resolving partition of the vertex set V(G) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in G are not contained in the same partition class. In this case, the coordinate of a vertex v in G is expressed in terms of the distances of v to all partition classes. This concept is a special case of the graph partition dimension notion. Previous authors have characterized all graphs of order n with locating-chromatic number either n or n-1. They also proved that there exists a tree of order n, n≥5, having locating-chromatic number k if and only if k âˆˆ{3,4,"¦,n-2,n}. In this paper, we characterize all trees of order n with locating-chromatic number n - t, for any integers n and t, where n > t+3 and 2 ≤ t < n/2

The Asymptotic value of the Locating-Chromatic number of nn-ary Tree

In 2013, Welyyanti et al. proved that $\chi_L(T(n,k))\leq n+k-1$ for all positive integers $n,k\geq2$. In this paper we prove that for any fixed integer $k\geq2$, almost all $n$-ary Tree $T(n,k)$ satisfy $\chi_L(T(n,k))=n+k-1$, moreover $\lim\limits_{n\to \infty} \chi_L(T(n,k))-n=k-1$.Comment: 4 Pages, 1 figur

The neighbor-locating-chromatic number of pseudotrees

Ak-coloringof a graphGis a partition of the vertices ofGintokindependent sets,which are calledcolors. Ak-coloring isneighbor-locatingif any two vertices belongingto the same color can be distinguished from each other by the colors of their respectiveneighbors. Theneighbor-locating chromatic number¿NL(G) is the minimum cardinalityof a neighbor-locating coloring ofG.In this paper, we determine the neighbor-locating chromatic number of paths, cycles,fans and wheels. Moreover, a procedure to construct a neighbor-locating coloring ofminimum cardinality for these families of graphs is given. We also obtain tight upperbounds on the order of trees and unicyclic graphs in terms of the neighbor-locatingchromatic number. Further partial results for trees are also established.Preprin

On locating-chromatic number of complete n-ary tree

Abstract Let c be a vertex k -coloring on a connected graph G(V, E) . Let Π = {C1, C2, ..., Ck} be the partition of V (G) induced by the coloring c . The color code cΠ(v) of a vertex v in G is (d(v, C1), d(v, C2), ..., d(v, Ck)), where d(v, Ci) = min{d(v, x)|x ∈ Ci} for 1 ≤ i ≤ k. If any two distinct vertices u, v in G satisfy that cΠ(u) = cΠ(v), then c is called a locating k-coloring of G . The locating-chromatic number of G, denoted by χL(G), is the smallest k such that G admits a locating k -coloring. Let T(n, k) be a complete n -ary tree, namely a rooted tree with depth k in which each vertex has n children except for its leaves. In this paper, we study the locating-chromatic number of T(n, k