Self-Dual Modular-Graceful Cyclic Digraphs

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 \( f(V) = \mathbb{Z}_{n+1}^* \), and \( \langle G^\wedge, f^\wedge \rangle \) is an isomorphic digraph of \( \langle G, f \rangle \). 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.