In a given graph G = (V, E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a c ≥ χ(G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by d(G, c). In this note we study d(G = Kn ×Kn, 2n−2). We determine an upper bound for d(G = Kn × Kn, 2n − 2) for all n and its exact value for some n.
