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Relationship magic between antimagic. (English) Zbl 1499.05555

Summary: Let \(G=(V(G),E(G))\) be a simple graph. A family \(\mathcal{H}=\{H_1,H_2,H_3,\dots,H_k\}\) of subgraphs of \(G\) is called an \(H\)-decomposition of \(G\) if every \(H_i\), \(1\leq i\leq k\), is isomorphic to \(H\), \(E(H_i)\cap E(H_j)=\emptyset\) for \(i\neq j\), and \(\bigcup^k_{i=1}E(H_i)=E(G)\). In such a case, we write \(G=H_1\oplus\cdots\oplus H_k\) and \(G\) is said to be \(H\)-decomposible.
An \((a,d)\)-\(H\)-antimagic decomposition of a graph \(G\) admitting an \(H\)-decomposition is a bijective function \(\varphi\colon V(G)\cup E(G)\to\{1,2,\dots,|V(G)|+|E(G)|\}\) such that for a subgraph \(H'\) isomorphic to \(H\), the \(H\)-weights \[\varphi(H')=\sum_{v\in V(H')}\varphi(v)+\sum_{e\in E(H')}\varphi(e)\] constitute an arithmetic progression \(a\), \(a+d\), \(a+2d,\dots,a+(k-1)d\) where \(a\) and \(d\) are positive integers and \(k\) is the number of all subgraphs of \(G\) isomorphic to \(H\). Such a labeling is called super if the smallest possible labels appear on the vertices and \(G\) is a super \((a,d)\)-\(H\)-antimagic decomposition graph. By using previous results on \((k,\delta)\)-anti balanced sets, we construct super \((a,d)\)-\(P_4\)-antimagic decomposition for prisms and we study the relationships between \((a,d)\)-\(H\)-magic decomposition and \((a,d)\)-\(H\)-antimagic decomposition.

MSC:

05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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