On super edge-magic decomposable graphs

Let G be any graph and let {Hi}i∈I be a family of graphs such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\left( {H_i } \right) \cap E\left( {H_j } \right) = \not 0$$\end{document} when i ≠ j, ∪i∈IE(Hi) = E(G) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\left( {H_i } \right) \ne \not 0$$\end{document} for all i ∈ I. In this paper we introduce the concept of {Hi}i∈I-super edge-magic decomposable graphs and {Hi}i∈I-super edge-magic labelings. We say that G is {Hi}i∈I-super edge-magic decomposable if there is a bijection β: V(G) → {1,2,..., |V(G)|} such that for each i ∈ I the subgraph Hi meets the following two requirements: β(V(Hi)) = {1,2,..., |V(Hi)|} and {β(a) +β(b): ab ∈ E(Hi)} is a set of consecutive integers. Such function β is called an {Hi}i∈I-super edge-magic labeling of G. We characterize the set of cycles Cn which are {H1, H2}-super edge-magic decomposable when both, H1 and H2 are isomorphic to (n/2)K2. New lines of research are also suggested.