[Paper Review] A Bound for the Locating Chromatic Numbers of Trees

Ali Behtoei, Mahdi Anbarloei|arXiv (Cornell University)|Aug 25, 2013

Graph Labeling and Dimension Problems5 references4 citations

TL;DR

This paper refutes a previously claimed bound on the locating chromatic number of trees by providing a counterexample and establishes a new, tight upper bound based on maximum degree. It constructs a family of trees with maximum degree $4 \times 3^{k-3}$ that achieve locating chromatic number $k$, proving the bound is best possible and correcting an error in prior work by Chartrand et al.

ABSTRACT

Let $f$ be a proper $k$-coloring of a connected graph $G$ and $Π=(V_1,V_2,\ldots,V_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $Π$ is defined to be the ordered $k$-tuple $c_{{}_Π}(v)=(d(v,V_1),d(v,V_2),\ldots,d(v,V_k)),$ where $d(v,V_i)=\min\{d(v,x): x\in V_i\}, 1\leq i\leq k$. If distinct vertices have distinct color codes, then $f$ is called a locating coloring. The minimum number of colors needed in a locating coloring of $G$ is the locating chromatic number of $G$, denoted by $\Cchi_{{}_L}(G)$. In this paper, we study the locating chromatic numbers of trees. We provide a counter example to a theorem of Gary Chartrand et al. [G. Chartrand, D. Erwin, M.A. Henning, P.J. Slater, P. Zhang, The locating-chromatic number of a graph, Bull. Inst. Combin. Appl. 36 (2002) 89-101] about the locating chromatic numbers of trees. Also, we offer a new bound for the locating chromatic number of trees. Then, by constructing a special family of trees, we show that this bound is best possible.

Motivation & Objective

• To identify and correct an error in a theorem by Chartrand et al. regarding the locating chromatic number of trees.
• To establish a new, tight upper bound for the locating chromatic number of trees in terms of their maximum degree.
• To construct a family of trees that achieve the proposed bound, proving it is best possible.
• To demonstrate that the previously claimed bound $\Delta(T) > (k-1)2^{k-2}$ is not tight for $k > 4$.
• To provide a constructive locating coloring for a family of trees that achieves the locating chromatic number $k$ with maximum degree $4 \times 3^{k-3}$.

Proposed method

• Constructs a counterexample tree $T_5$ with maximum degree 36 and locating chromatic number 5, contradicting the claim that $\Delta(T) > 32$ implies $\chi_L(T) \geq 6$.
• Introduces a general family of trees $T_k$ for $k \geq 5$ with vertex set $V(T_k)$ and edge set $E(T_k)$ defined via recursive structure based on binary and ternary representations of indices.
• Defines a vertex $k$-coloring $f_k$ assigning colors based on position and index values, with distinct color classes $V_i$.
• Computes color codes $c_\Pi(v)$ as $k$-tuples of distances to each color class, using Kronecker deltas and vector operations.
• Uses vector notation $e_i$ and $e = \sum e_i$ to express color codes systematically, ensuring distinctness across vertices.
• Proves that all color codes are distinct by analyzing the structure of $\alpha_i$ and $\bar{\alpha}_i$, showing no two vertices share the same code.

Experimental results

Research questions

• RQ1 Is the bound $\Delta(T) > (k-1)2^{k-2}$ for $\chi_L(T) > k$ valid for all $k \geq 3$?
• RQ2 Can a tree with maximum degree greater than $(k-1)2^{k-2}$ have locating chromatic number $k$?
• RQ3 Is the previously claimed bound for the locating chromatic number of trees tight?
• RQ4 What is the best possible upper bound for the locating chromatic number of a tree in terms of its maximum degree?
• RQ5 Can a constructive family of trees be built to achieve the optimal locating chromatic number for a given maximum degree?

Key findings

• A counterexample is provided: a tree $T_5$ with maximum degree 36 has locating chromatic number exactly 5, contradicting the claim that $\Delta(T) > 32$ implies $\chi_L(T) \geq 6$.
• The new bound $\chi_L(T) \leq k$ holds for trees with maximum degree $\Delta(T) \leq 4 \times 3^{k-3}$, and this bound is tight.
• The family of trees $T_k$ achieves $\chi_L(T_k) = k$ and $\Delta(T_k) = 4 \times 3^{k-3}$, proving the bound is best possible.
• The locating coloring $f_k$ is explicitly constructed and proven to assign distinct color codes to all vertices, confirming it is a valid locating coloring.
• The color code of each vertex is uniquely determined by its position and index, with vector-based expressions ensuring no code collisions.
• The result shows that the earlier bound by Chartrand et al. is not tight for $k > 4$, and the new bound $4 \times 3^{k-3}$ is optimal for achieving $\chi_L(T) = k$.