(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let H_1 and H_2 be two non-isomorphic graphs. A graph G is said to admit an (H_1,H_2)-covering if every edge of G is contained in either a subgraph of G isomorphic to H_1 or to H_2. We say that a graph G admitting an (H_1,H_2)-covering is (H_1,H_2)-magic if there exists a total labeling f : V(G) E(G) → [1,|V(G)|+|E(G)|] such that there exist magic constants c_1 and c_2 such that the weight of every subgraph H_i^* of G isomorphic to H_i equals to c_i, i=1, 2. The weight of a subgraph H is defined as align* w(H) = _v V(H) f(v) + _e E(H) f(e). align* Moreover, a graph G is called (H_1,H_2)-supermagic if the vertices are labeled with the numbers from 1 up to |V(G)|. In this paper, we present some constructions of (H_1,H_2)-magic graphs.