An H-super magic decompositions of the lexicographic product of graphs
H. Hendy, Kiki Ariyanti Sugeng, A.N.M. Salman · AIP conference proceedings · 2018
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 Kn,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).