Cycle-magic graphs

A simple graph G=(V,E)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|}f:V∪E→{1,2,…,|V|+|E|} such that, for every subgraph H′=(V′,E′)H′=(V′,E′) of G isomorphic to C  , ∑v∈V′f(v)+∑e∈E′f(e)∑v∈V′f(v)+∑e∈E′f(e) is constant. When f(V)={1,…,|V|}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 CrCr-magic graphs for each r⩾3r⩾3. The results rely on a technique of partitioning sets of integers with special properties.