An H-super magic decompositions of the lexicographic product of graphs

Let H and G be two simple graphs. The topic of an H-magic decomposition of G arises from the combination of graph decomposition and graph labeling. A decomposition of a graph G into isomorphic copies of a graph H is H - magic if there is a bijection f: V(G) ∪ E(G) → {1,2, …, |V(G) ∪ E(G)|} such that the sum of labels of edges and vertices of each copy of H in the decomposition is constant. A lexicographic product of two graphs G₁ and G₂, denoted by G₁ [G₂], is a graph which arises from G₁ by replacing each vertex of G₁ by a copy of the G₂ and each edge of G₁ by all edges of the complete bipartite graph K_n,n where n is the order of G₂, In this paper we show that for n ≥ 4 and m ≥ 2, the lexicographic product of the cycle graphs complement and complete graphs complement $Cn¯[Km¯]$ has $P2[Km¯]$- magic decomposition if and only if m is even, or m is odd and n ≡ 1 (mod4), or m is odd and n ≡ 2 (mod4).