Constructions of \(H\)-Antimagic Graphs Using Smaller Edge-Antimagic Graphs

Abstract

A simple graph \(G = (V, E)\) admits an \(H\)-covering if every edge in \(E\) belongs to at least one subgraph of \(G\) isomorphic to a given graph \(H\). An \((a, d)\)-\(H\)-antimagic labeling of \(G\) admitting an \(H\)-covering is a bijective function \(f : V \cup E \rightarrow \{1, 2, \ldots, |V| + |E|\}\) such that, for all subgraphs \(H’\) of \(G\) isomorphic to \(H\), the \(H’\)-weights, \(wt(H’) = \sum_{v \in V(H’)} f(v) + \sum_{e \in E(H’)} f(e)\), constitute an arithmetic progression with the initial term \(a\) and the common difference \(d\). Such a labeling is called super if \(f(V) = \{1, 2, \ldots, |V|\}\). In this paper, we study the existence of super \((a, d)\)-\(H\)-antimagic labelings for graph operation \(G ^ H\), where \(G\) is a (super) \((b, d^*)\)-edge-antimagic total graph and \(H\) is a connected graph of order at least \(3\).