On (H1,H2)-magic generalized total composition

Abstract

Let $H_1$ and $H_2$ be two non-isomorphic graphs. A graph $G$ is said to admit an \textit{$(H_1,H_2)$-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 \textit{$(H_1,H_2)$-magic} if there exists a total labeling $f : V(G) \cup E(G) \to [1,|V(G)|+|E(G)|]$ 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 
\begin{align*}
    w(H) = \sum_{v \in V(H)} f(v) + \sum_{e \in E(H)} f(e).
\end{align*}
Moreover, a graph $G$ is called \textit{$(H_1,H_2)$-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.