NOTE ON SUPER \((a,1)\)–\(P_3\)–ANTIMAGIC TOTAL LABELING OF STAR \(S_n\)

Abstract

Let \(G=(V, E)\) be a simple graph and \(H\) be a subgraph of \(G\). Then \(G\) admits an \(H\)-covering, if every edge in \(E(G)\) belongs to at least one subgraph of \(G\) that is isomorphic to \(H\). An \((a,d)-H\)-antimagic total labeling of \(G\) is bijection \(f:V(G)\cup E(G)\rightarrow \{1, 2, 3,\dots, |V(G)| + |E(G)|\}\) such that for all subgraphs \(H'\) of \(G\) isomorphic to \(H\), the \(H'\) weights \(w(H') =\sum_{v\in V(H')} f (v) + \sum_{e\in E(H')} f (e)\) constitute an arithmetic progression \(\{a, a + d, a + 2d, \dots , a + (n- 1)d\}\), where \(a\) and \(d\) are positive integers and \(n\) is the number of subgraphs of \(G\) isomorphic to \(H\). The labeling \(f\) is called a super \((a, d)-H\)-antimagic total labeling if \(f(V(G))=\{1, 2, 3,\dots, |V(G)|\}.\) In [5], David Laurence and Kathiresan posed a problem that characterizes the super \( (a, 1)-P_{3}\)-antimagic total labeling of Star \(S_{n},\) where \(n=6,7,8,9.\)  In this paper, we completely solved this problem.