Super (a, 3)-edge Antimagic Total Labeling for Union of Two Stars

Abstract

An (a,d)-edge antimagic total labeling of a (p, q)-graph G is bijection $f:V\cup E\rightarrow \{1,2,3,\dots ,p+q\}$ with the property that the edge-weights $w(uv)=f(u)+f(v)+f(uv)$ where $uv\in E(G)$ form an arithmetic progression $a, a+d,\dots ,a+(q-1)d$, where $a>0$ and $d\ge 0$ are two fixed integers. If such a labeling exists, then G is called an (a,d)-edge antimagic total graph. If further the vertex labels are the integers $\{1,2,3,\dots ,p\}$, then f is called a super (a,d)-edge antimagic total labeling of G ((a, d)-SEAMT labeling) and a graph which admits such a labeling is called a super (a,d)-edge antimagic total graph ((a, d)-SEAMT graph). If $d=0,$ then the graph G is called a super edge-magic graph. In this paper we investigate the existence of super (a, 3)-edge antimagic total labelings for union of two stars.

S. Arumugam—Also at School of Electrical Engineering and Computer Science, The University of Newcastle, NSW 2308, Australia; Department of Computer Science, Liverpool Hope University, Liverpool, UK; Department of Computer Science, Ball State University, USA.