On Super \((a, 2)\)-Edge-Antimagic Total Labeling of Disconnected Graphs

Abstract

A labeling of a graph is a mapping that carries some set of graph elements into numbers (usually the positive integers). An \((a, d)\)-edge-antimagic total labeling of a graph with \(p\) vertices and \(q\) edges is a one-to-one mapping that takes the vertices and edges onto the integers \(1, 2, \ldots, p + q\), such that the sums of the label on the edges and the labels of their end points form an arithmetic sequence starting from \(a\) and having a common difference \(d\). Such a labeling is called \({super}\) if the smallest possible labels appear on the vertices. In this paper, we study the super \((a, 2)\)-edge-antimagic total labelings of disconnected graphs. We also present some necessary conditions for the existence of \((a, d)\)-edge-antimagic total labelings for \(d\) even.