The locating chromatic number of the join of graphs

Abstract

‎Let $f$ be a proper $k$-coloring of a connected graph $G$ and‎ <br />‎$Pi=(V_1,V_2,ldots,V_k)$ be an ordered partition of $V(G)$ into‎ <br />‎the resulting color classes‎. ‎For a vertex $v$ of $G$‎, ‎the color‎ <br />‎code of $v$ with respect to $Pi$ is defined to be the ordered‎ <br />‎$k$-tuple $c_{{}_Pi}(v)=(d(v,V_1),d(v,V_2),ldots,d(v,V_k))$‎, <br />‎where $d(v,V_i)=min{d(v,x):~xin V_i}‎, ‎1leq ileq k$‎. ‎If‎ <br />‎distinct vertices have distinct color codes‎, ‎then $f$ is called a‎ <br />‎locating coloring‎. ‎The minimum number of colors needed in a‎ <br />‎locating coloring of $G$ is the locating chromatic number of $G$‎, <br />‎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 $Cchi_{{}_L}(G_1vee G_2)=Cchi_{{}_L}(G_1)+Cchi_{{}_L}(G_2)$‎, ‎where $G_1vee G_2$ is the join of $G_1$ and $G_2$‎. ‎Also‎, ‎we determine the‎ <br />‎locating chromatic number of the join of paths‎, ‎cycles and complete multipartite graphs‎.