Self-Dual Modular-Graceful Cyclic Digraphs

Abstract

In this paper, we introduce, for the first time, the notion of self-dual modular-graceful labeling of a cyclic digraph. A cyclic digraph $G(V,E)$ is a digraph whose connected components are directed cycles. The line digraph $G^{\wedge }(V^{\wedge },E^{\wedge })$ of the cyclic digraph $G$ is the digraph where $V^{\wedge }=E$, $E^{\wedge }=V$, and if $\alpha ,\beta$ are two edges of $G$ which join vertex $x$ to vertex $y$ and vertex $y$ to vertex $z$ respectively, then in the digraph $G^{\wedge }$, $y$ is the edge joining vertex $\alpha$ to vertex $\beta$. A labeling $f$ for a cyclic digraph of order $n$ is a map from $V$ to ${\mathbb{Z}}_{n+1}$. The labeling $f$ induces a dual labeling $f^{\wedge }$ for $G^{\wedge }$ by $f^{\wedge }(\alpha )=f(x)–f(y)$, where $\alpha$ is an edge of $G$ which joins vertex $x$ to vertex $y$. A self-dual modular-graceful cyclic digraph $G$ is a cyclic digraph together with a labeling $f$ where the image $f(V)={\mathbb{Z}}_{n+1}^{\ast }$, and $⟨G^{\wedge },f^{\wedge }⟩$ is an isomorphic digraph of $⟨G,f⟩$. We prove the necessary and sufficient conditions for the existence of self-dual modular-graceful cyclic digraphs and connected self-dual modular-graceful cyclic digraphs. We also give some explicit constructions of these digraphs in the case $n+1$ is prime and in the general case where $n+1$ is not prime.