Graceful Labelings of Directed Graphs

A graceful labeling of a directed graph \( D \) with \( e \) edges is a one-to-one map \( \theta: V(D) \to \{0, 1, \dots, e\} \) such that \( \theta(y) – \theta(x) \mod (e + 1) \) is distinct for each \( (x, y) \in E(D) \). This paper summarizes previously known results on graceful directed graphs and presents some new results on directed paths, stars, wheels, and umbrellas.