On H-supermagic Coverings Of Two Classes Graphs

Suppose a simple graph G=(V(G), E(G)), a covering of G is a family of subgraphs, say B={Ⅱ1,..., Ⅱk},having the property that each edge of G is contained in at least one graph Hi, for some i. If all Ⅱi,1≤i≤k, are isomorphic to graph H, such a covering is called an Ⅱ—covering of G.The graph G is H-magic if there exists a bijection f:V(G)(?)E(G)�'[1,|V(G)|+|E(G)|] such that for every subgraph Ⅱ∈B,∑u∈V(H) f(u)+∑e∈E(H)f(e) is constant, then f is said to be an Ⅱ-magic labeling of G. In particular, G is Ⅱ-supermagic if f(V(G))=[1,|V(G)|], then∫is said to be an H-supermagic labeling of G.The supermagic labeling is a new concept which connects graph labeling with graph cover-ing. It plays an important role in the field of graph theory. The supermagic labeling is concerned much more by scholars who are at home and abroad. In this paper,we use the methods of algebra and combinatorics to study supermagic labelings.The paper is divided into three sections. In section1, we introduce the research back-ground, basic concept and the known results about graph labelings. In section2, we use the methods of direct construction and recursive structure to obtain some set partition results with restricted conditions. In section3, the following theorems can be obtained by applying the results in section2:●For any integer k≥1, m≥3, n≥1, t≥0, Pk·(Cm+t)n is (Cm+t)-supermagic.●For any integer m≥2,n≥2, Pm×Pn is C4-supermagic.