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On (F,H)-sim-magic labelings of graphs

Abstract:

A simple graph G(V,E) admits an H-covering if every edge in G belongs to a subgraph of G isomorphic to H. In this case, G is called H-magic if there exists a bijective function f: V ∪ E → {1, 2,…, |V | + |E|}, such that for every subgraph H′ of G isomorphic to H, (Formula Presented) is constant. Moreover, G is called H-supermagic if f: V (G) → {1, 2,…, |V |}. This paper generalizes the previous labeling by introducing the (F,H)-sim-(super) magic labeling. A graph admitting an F-covering and an H-covering is called (F,H)-sim-(super) magic if there exists a function f that is F-(super)magic and H-(super)magic at the same time. We consider such labelings for two product graphs: the join product and the Cartesian product. In particular, we establish a sufficient condition for the join product G+H to be (K2+H, 2K2+H)- sim-supermagic and show that the Cartesian product G × K2 is (C4,H)-sim-supermagic, for H isomorphic to a ladder or an even cycle. Moreover, we also present a connection between an α-labeling of a tree T and a (C4,C6)-sim-supermagic labeling of the Cartesian product T × K2