Edge-Antimagicness for a Class of Disconnected Graphs

Abstract

Suppose \(G\) is a finite graph with vertex-set \(V(G)\) and edge-set \(E(G)\). An \((a, d)\)-edge-antimagic total labeling on \(G\) is a one-to-one map \(f\) from \(V(G) \cup E(G)\) onto the integers \(1, 2, \ldots, |V(G)| + |E(G)|\) with the property that the edge-weights \(w(uv) = f(u) + f(v) + f(uv)\), \(uv \in E(G)\), form an arithmetic progression starting from \(a\) and having common difference \(d\). Such a labeling is called super if the smallest labels appear on the vertices. In this paper, we investigate the existence of super \((a, d)\)-edge-antimagic total labelings of disjoint union of multiple copies of complete bipartite graph.