On (F, H)-simultaneously-magic labelings of graphs - Yeva Fadhilah Ashari; A.N.M. Salman; Rinovia Simanjuntak; Andrea Semaničová-Feňovčíková; Martin Baca (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2023)
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, wtf(H′) = Σv ∈ V(H′)f(v)+ Σe ∈ 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 (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.