# The Locating Chromatic Number of the Join of Graphs

Research paper by **Ali Behtoei**

Indexed on: **21 Jan '12**Published on: **21 Jan '12**Published in: **Mathematics - Combinatorics**

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#### 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.