The Locating Chromatic Number of the Join of Graphs

TL;DR

This paper investigates the locating chromatic number of graph joins, establishing formulas for connected graphs with diameter at most two and computing values for joins of paths, cycles, and multipartite graphs.

Contribution

It introduces a formula for the locating chromatic number of joins of certain graphs and computes this number for specific graph classes.

Findings

For graphs with diameter at most two, the locating chromatic number of their join equals the sum of their individual locating chromatic numbers.

The paper determines the locating chromatic number for joins of paths, cycles, and complete multipartite graphs.

Provides new insights into the structure of locating colorings in graph joins.

Abstract

Let $f$ be a proper $k$-coloring of a connected graph $G$ and $\Pi=(V_1,V_2,...,V_k)$ 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 $\Pi$ is defined to be the ordered $k$-tuple $c_{{}_\Pi}(v):=(d(v,V_1),d(v,V_2),...,d(v,V_k)),$ 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 number of the join of graphs. We show that when $G_1$ and $G_2$ are two connected graphs with diameter at most two, then $\Cchi_{{}_L}(G_1+G_2)=\Cchi_{{}_L}(G_1)+\Cchi_{{}_L}(G_2)$, where $G_1+G_2$ is the join of $G_1$ and $G_2$. Also, we determine the locating chromatic numbers of the join of paths, cycles and complete multipartite graphs.