The Neighbor-Locating-Chromatic Number of Pseudotrees

Abstract

A $k$-coloring of a graph $G$ is a partition of the set of vertices of $G$ into $k$ independent sets, which are called colors. A $k$-coloring is neighbor-locating if any two vertices belonging to the same color can be distinguished from each other by the colors of their respective neighbors. The neighbor-locating chromatic number $χ_{_{NL}}(G)$ is the minimum cardinality of a neighbor-locating coloring of $G$. 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 of minimum cardinality for these families of graphs is given. We also obtain tight upper bounds on the order of trees and unicyclic graphs in terms of the neighbor-locating chromatic number. Further partial results for trees are also established.

Figures (8)

• Figure 1: From left to right, a 3-NL-coloring of cycles $C_{3}$, $C_{5}$, $C_{7}$ and $C_{9}$. A 3-NL-coloring of paths $P_{4}$, $P_{6}$ and $P_{8}$ can be obtained by removing the squared vertices and a 3-NL-coloring of paths $P_{3}$, $P_{5}$, $P_{7}$ and $P_{9}$ can be obtained by removing the edges $a$, $b$, $c$ and $d$, respectively.
• Figure 2: Illustrating Definition \ref{['pro.insertarpar']}. Left, $|\{i,j,h\}|=3$ and right, $|\{i,j,a,b\}|=4$.
• Figure 3: Obtaining a 4-NL-coloring from a 1-paired $3$-NL-coloring of the cycle $C_9$. In white, the vertices of color-degree 1. Recall that $\ell(3)=9$, $\ell(4)=24$, $a_2(4)-1=11$ and $a_2(4)=12$. Inserting vertices of color-degree 2 in some edges of $C_{9}$, we achieve 4-NL-colorings of $C_{n}$, whenever $n\in \{10,11,12\}$. Inserting pairs of white vertices in some edges of $C_{11}$ and of $C_{12}$ we achieve 4-NL-colorings of $C_n$, whenever $n\in \{13,\dots ,24\}\setminus \{23\}$.
• Figure 4: A 5-NL-coloring of the comb $B_{20}$, a 6-NL-coloring of the comb $B_{30}$ and a 7-NL-coloring of the comb $B_{42}$. In white, adjacent vertices of $M_r$ with no consecutive colors modulo $k$. In $B_{20}$ and in $B_{42}$, we have shifted the colors of the vertices in gray with respect to the general rule used to the vertices of $M_r$, when $r$ is odd. Below, the general rule for coloring adjacent vertices of consecutive groups $M_r$ and $M_{r+1}$ and the leaves hanging from them. In all cases, the colors involved are $r-1$, $r$, $r+1$ and $r+2$.
• Figure 5: A 6-NL-coloring of the unicyclic graph $U_6$.

Theorems & Definitions (50)

• Remark \oldthetheorem
• Theorem \oldthetheorem: xnl1
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• Corollary \oldthetheorem
• proof
• Remark \oldthetheorem
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