The Dimension-Balanced Hamiltonian Cycle Problem (DBH problem) has been widely studied due to its practical relevance. In applications where edge direction matters, the DBH problem naturally extends to directed graphs. We introduce the Directed Dimension-Balanced Hamiltonian Cycle Problem (DDBH problem) as this extension. Consider an undirected graph G represented as a symmetric digraph D, where each undirected edge corresponds to two symmetric directed edges. Let Ẽ be a partition of the edge set of G into k disjoint subsets, which is then used to construct a partition F̃ of the edge set of D into 2k disjoint subsets. A DDBH on D with respect to F̃ is a directed Hamiltonian cycle in which the number of edges from any two subsets of F̃ differs by at most one. We study the existence of DDBH in toroidal mesh digraphs DTm,n for m, n ≥ 3. We show nonexistence when both m and n are odd or either is 3, and prove existence in all other cases when mn is not divisible by 8.
Juan et al. (Mon,) studied this question.