A graph
G admits an
H-covering if every edge of
G belongs to a subgraph isomorphic to a given graph
H.
G is said to be
H-magic if there exists a bijection
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A graph
G admits an
H-covering if every edge of
G belongs to a subgraph isomorphic to a given graph
H.
G is said to be
H-magic if there exists a bijection
such that
is a constant, for every subgraph
isomorphic to
H. In particular,
G is said to be
H-supermagic if
. When
H is isomorphic to a complete graph
, an
H-(super)magic labeling is an edge-(super)magic labeling. Suppose that
G admits an
F-covering and
H-covering for two given graphs
F and
H. We define
G to be
-sim-(super)magic if there exists a bijection
that is simultaneously
F-(super)magic and
H-(super)magic. In this paper, we consider
-sim-(super)magic where
H is isomorphic to three classes of graphs with varied symmetry: a cycle which is symmetric (both vertex-transitive and edge-transitive), a star which is edge-transitive but not vertex-transitive, and a path which is neither vertex-transitive nor edge-transitive. We discover forbidden subgraphs for the existence of
-sim-(super)magic graphs and classify classes of
-sim-(super)magic graphs. We also derive sufficient conditions for edge-(super)magic graphs to be
-sim-(super)magic and utilize such conditions to characterize some
-sim-(super)magic graphs.
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