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 $1,2,\dots ,p+q$, 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.