Defining sets in (proper) vertex colorings of the Cartesian product of a cycle with a complete graph

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). The d(G = Cm × Kn, χ(G)) has been studied. In this note we show that the exact value of defining number d(G = Cm × Kn, c) with c > χ(G), where n ≥ 2 and m ≥ 3, unless the defining number d(K3 × C2r, 4), which is given an upper and lower bounds for this defining number. Also some bounds of defining number are introduced.