Coloring digraphs with forbidden cycles

Let k and r be two integers with k >= 2 and k >= 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 Is 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 not equal 2 and (k+1)-colorable otherwise. Our results also strengthen several classical theorems on graph coloring proved by Bondy, Erdos and Hajnal, Gallai and Roy, Gyarffis, etc. (C) 2015 Elsevier Inc. All rights reserved.