A simple graph G=(V,E) admits a cycle-covering if every edge in E belongs at least to one subgraph of G isomorphic to a given cycle C. Then the graph G is C-magic if there exists a total labelling f : V ∪ E → {1, 2, . . . , |V | + |E|} such that, for every subgraph H'=(V',E') of G isomorphic to C, $\Sigma_{v\in V'^{f{(v)}}}$ + $\Sigma{e \in E'}f^{(e)}$ is constant. When f(V)= {1, . . . , |V|}, then G is said to be C-supermagic. We study the cyclic-magic and cyclic-supermagic behavior of several classes of connected graphs. We give several families of Cr -magic graphs for each r≥3. The results rely on a technique of partitioning sets of integers with special properties.