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 \(π=\{S₁,S₂,…,Sₖ\}\) be an ordered partition of the vertex set \(V(G)\) into the resulting color classes, where \(Sᵢ\) is the set of all vertices that receive the color \(i\). For a vertex \(v\) of \(G\), the color code \(cπ(v)\) of \(v\) with respect to \(π\) is the ordered \(k\)-tuple \(cπ(v)=(d(v,S₁),d(v,S₂),…,d(v,Sₖ))\), where \(d(v,Sᵢ)=min\{d(v,u): u∈ Sᵢ\}\) 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.