Magic Coverings

A simple graph \( G = (V,E) \) admits an \( H \)-\emph{covering} if every edge in \( E \) belongs to a subgraph of \( G \) isomorphic to \( H \). In this case, we say that \( G \) is \( H \)-\emph{magic} if there is a total labeling \( f : V \cup E \rightarrow \{1,2,\ldots,|V|+|E|\} \) such that for each subgraph \( H’ = (V’,E’) \) of \( G \) isomorphic to \( H \),

\[
\sum_{v \in V’} f(v) + \sum_{e \in E’} f(e)
\]

is constant. When \( f(V) = \{1,\ldots,|V|\} \), we say that \( G \) is \( H \)-\emph{supermagic}.

We study \( H \)-magic graphs for several classes of connected graphs. We also provide constructions of infinite families of \( H \)-magic graphs for an arbitrary given graph \( H \).