Coloring digraphs with forbidden cycles

Let k and r   be two integers with k≥2k≥2 and k≥r≥1k≥r≥1. In this paper we show that (1) if a strongly connected digraph D contains no directed cycle of length 1 modulo k, then D is k-colorable; and (2) if a digraph D contains no directed cycle of length r modulo k, then D can be vertex-colored with k colors so that each color class induces an acyclic subdigraph in D. Our results give affirmative answers to two questions posed by Tuza in 1992. Moreover, the second one implies the following strong form of a conjecture of Diwan, Kenkre and Vishwanathan: If an undirected graph G contains no cycle of length r modulo k, then G is k  -colorable if r≠2r≠2 and (k+1)(k+1)-colorable otherwise. Our results also strengthen several classical theorems on graph coloring proved by Bondy, Erdős and Hajnal, Gallai and Roy, Gyárfás, etc.