A New Kind of Dominated Coloring of Some Special Graphs

This paper introduces the concept of locating-dominated coloring, a new graph coloring parameter that merges the properties of dominated coloring and locating coloring. For a connected graph G, a locating-dominated coloring is defined as a proper dominated k-coloring of G using an ordered partition of VG to k-color classes Π=C1,C2,…,Ck such that for every two distinct vertices x and y, we have cΠx≠cΠy, where cΠx=dx,C1,dx,C2,⋯,dx,Ck and dx,Ci=mindx,t; t∈Ci. The primary objective is to investigate this new coloring parameter, determine its exact values for various graph families, for instance, paths, cycles, complete graphs, unicycle, Helm, and some more graphs, as well as for Cartesian products including Pm□Pn, Pm□Cn, Pm□Kn, and Km□Kn, and compare it with existing coloring parameters. The paper concludes with a discussion of the advantages and limitations of this new coloring, along with open problems for future research.