V- Super vertex in-antimagic total graceful labelings of digraphs

Abstract

Let $D = (V, A)$ be a directed graph with $p$ vertices and $q$ arcs. A $V$-super vertex in-antimagic total graceful labeling ($V$-SVIAMTG labeling) is a bijection $f: V(D) \cup A(D) \rightarrow \{ 1,2 , \ldots, \\p + q \}$ with the property that $f(V(D)) = \{ 1,2,\ldots, p \}$ and for each $v \in V(D)$, $|w(v)-f(v)| $ consists of distinct integers, where $w(v)=\sum\limits_{u \in I(v)} f((u,v))$. A digraph $D$ is called a $V$-super vertex in-antimagic total graceful digraph ($V$-SVIAMTG digraph) if $D$ admits a $V$-SVIAMTG labeling. In this paper, we introduce and study the existence of $V$-SVIAMTG for some classes of digraphs