On Connection Between $\alpha$-Labelings and Edge-Antimagic Labelings of Disconnected Graphs

Abstract

A labeling of a graph is any map that carries some set of graph elements to numbers (usually to the positive integers). An $(a,d)$-edge-antimagic total labeling on a graph with $p$ vertices and $q$ edges is defined as a one-to-one map taking the vertices and edges onto the integers $1,2,\dots ,p+q$ with the property that the sums of the labels on the edges and the labels of their endpoints 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.

We use the connection between $a$-labelings and edge-antimagic labelings for determining a super $(a,d)$-edge-antimagic total labelings of disconnected graphs.