A lower bound for the packing chromatic number of the Cartesian product of cycles

Let G = (V, E) be a simple graph of order n and i be an integer with i ≥ 1. The set X_i ⊆ V(G) is called an i-packing if each two distinct vertices in X_i are more than i apart. A packing colouring of G is a partition X = {X₁, X₂,..., X_k} of V(G) such that each colour class X_i is an i-packing. The minimum order k of a packing colouring is called the packing chromatic number of G, denoted by χ_ρ(G). In this paper we show, using a theoretical proof, that if q = 4t, for some integer t ≥ 3, then 9 ≤ χ_ρ(C₄ □ C_q). We will also show that if t is a multiple of four, then χ_ρ(C₄ □ C_q) = 9.