Distance-2 Chromatic Number of the Central and Shadow Graphs of a Cycle and its Application to Computer Science

Let \(G=(V(G),E(G))\) be a graph. The chromatic number of a graph \(G\), \(\chi(G)\), is the minimum number of colors needed to color the vertices such that no two adjacent vertices share the same color. The distance-2 chromatic number of G, \(\chi_2 (G)\), extends this by ensuring that no two vertices within distance 2 share the same color in G. This study investigates \(\chi_2 (G)\) for the Shadow and Central graphs of cycles. While previous research has focused on certain graphs, we expand the analysis to the Shadow and Central graphs of cycles, with potential applications in computer science, particularly in network topology and resource allocation.