The locating chromatic number of the join of graphs

‎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.