Locating Chromatic Number of Middle Graph of Path, Cycle, Star, Wheel, Gear and Helm Graphs

Let \ (c\) be a proper \ (k\) -coloring of a connected graph \ (G\) and \ (=\S₁, S₂, , S₊\\) be an ordered partition of the vertex set \ (V (G) \) into the resulting color classes, where \ (S₈\) is the set of all vertices that receive the color \ (i\). For a vertex \ (v\) of \ (G\), the color code \ (c_ (v) \) of \ (v\) with respect to \ (\) is the ordered \ (k\) -tuple \ (c_ (v) = (d (v, S₁), d (v, S₂), , d (v, S₊) ) \), where \ (d (v, S₈) =min\d (v, u): u S₈\\) for \ (1 i k\). If all distinct vertices of \ (G\) have different color codes, then \ (c\) is called a locating coloring of \ (G\). The locating chromatic number is the minimum number of colors needed in a locating coloring. In this paper, we determine the locating-chromatic number for the middle graphs of Path, Cycle, Wheel, Star, Gear and Helm graphs.