Tree-antimagicness of disconnected graphs (Q1665840)

Summary: A simple graph \(G\) admits an \(H\)-covering if every edge in \(E(G)\) belongs to a subgraph of \(G\) isomorphic to \(H\). The graph \(G\) is said to be (\(a\), \(d\))-\(H\)-antimagic if there exists a bijection from the vertex set \(V(G)\) and the edge set \(E(G)\) onto the set of integers \(\left\{1, 2, \ldots, \left|V \left(G\right)\right| + \left|E(G)\right|\right\}\) such that, for all subgraphs \(H^\prime\) of \(G\) isomorphic to \(H\), the sum of labels of all vertices and edges belonging to \(H^\prime\) constitute the arithmetic progression with the initial term \(a\) and the common difference \(d\). \(G\) is said to be a super (\(a\), \(d\))-\(H\)-antimagic if the smallest possible labels appear on the vertices. In this paper, we study super tree-antimagic total labelings of disjoint union of graphs.