On (F, H)-simultaneously-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 , w t f ( H ′) = Σ v ∈ V ( H ′) f ( v )+ Σ e ∈ E ( H ′) f ( e ) 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 ( K 2 + H , 2 K 2 + H ) -sim-supermagic and show that the Cartesian product G × K 2 is ( C 4 , 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 ( C 4 , C 6 ) -sim-supermagic labeling of the Cartesian product T × K 2 .