Kernels by monochromatic paths in digraphs with covering number 2

We call the digraph D an m-colored digraph if the arcs of D are colored with m colors. A subdigraph H of D is called monochromatic if all of its arcs are colored alike. A set N ⊆ V (D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u, v ∈ N there is no monochromatic directed path between them and; (ii) for every vertex x ∈ (V (D)\N) there is a vertex y ∈ N such that there is an xy-monochromatic directed path. The covering number of an m-colored digraph D, denoted o(D) is the minimum number of transitive tournaments of D that partition V (D) In this paper we prove that if D is an m-colored digraph with o(D) = 2 such that every directed cycle of length 3, 4 or 5 is monochromatic, then D has a kernel by monochromatic paths. This result gives a positive answer (for digraphs with o(D) = 2) of the following question which has motivated many results in monochromatic kernel theory: Given a digraph D is there an integer k such that if every directed cycle of length at most k is monochromatic, then D has a kernel by monochromatic paths?.