Edge magic total labeling of lexicographic product C4(2r+1) o ~K2 cycle with chords, unions of paths, and unions of cycles and paths

An edge magic total (EMT) labeling of a graph G = (V, E) is a bijection from the set of vertices and edges to a set of numbers defined by λ : V ∪ E → {1, 2, ..., ∣V∣ + ∣E∣} with the property that for every xy ∈ E, the weight of xy equals to a constant k, that is, λ(x) + λ(y) + λ(xy) = k for some integer k. In this paper given the construction of an EMT labeling for certain lexicographic product $C_{4(2r+1)}\circ \overline{K_2}$, cycle with chords ^[c]tC_n, unions of paths mP_n, and unions of cycles and paths m(C_{n1(2r + 1)} ∪ (2r + 1)P_n2).