Spanning Directed Trails in Semicomplete 3-Multipartite Digraphs

The study of spanning trails in digraphs associates two fundamental concepts: connectivity and independence. For general digraphs, establishing universal conditions remains an open challenge, but semi-complete multipartite digraphs reveal precise connectivity restrictions due to their well-defined structure. In this work, we prove that every strongly connected semi-complete 3-multipartite digraph D with arc-strong connectivity $\lambda(D) \ge \alpha(D)-1$ contains a spanning directed trail. This result provides evidence that multipartite structures may require weaker connectivity conditions than arbitrary digraphs, opening new directions for studying spanning trails in constrained digraph classes.