Author/Authors :
Behtoei، Ali نويسنده Isfahan University of Technology , , Anbarloei، Mahdi نويسنده Imam Khomeini International University ,
Abstract :
Let $f$ be a proper $k$-coloring of a connected graph $G$ and
$\Pi=(V_1,V_2,\ldots,V_k)$ 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 $\Pi$ is defined to be the ordered
$k$-tuple $c_{{}_\Pi}(v)=(d(v,V_1),d(v,V_2),\ldots,d(v,V_k))$,
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 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 $\Cchi_{{}_L}(G_1\vee G_2)=\Cchi_{{}_L}(G_1)+\Cchi_{{}_L}(G_2)$, where $G_1\vee G_2$ is the join of $G_1$ and $G_2$. Also, we determine the
locating chromatic number of the join of paths, cycles and complete multipartite graphs.
