Note on super $(a,1)-P_{3}$-antimagic total labeling of star $S_n$

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.