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 C_n[K_m] has P₂[K_m]-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), or m is odd and n ≡ 2(mod4) . [ABSTRACT FROM AUTHOR]