Kernels and cycles' subdivisions in arc-colored tournaments

Let D be a digraph. D is said to be an m-colored digraph if the arcs of D are colored with m colors. A path P in D is called monochromatic if all of its arcs are colored alike. Let D be an m-colored digraph. A set N ⊆ V (D) is said to be a kernel by monochromatic paths of D if it satisfies the following conditions: a) for every pair of different vertices u, v ∈ N there is no monochromatic directed path between them; and b) for every vertex x ∈ V (D) − N there is a vertex n ∈ N such that there is an xn-monochromatic directed path in D. In this paper we prove that if T is an arc-colored tournament which does not contain certain subdivisions of cycles then it possesses a kernel by monochromatic paths. These results generalize a well known sufficient condition for the existence of a kernel by monochromatic paths obtained by Shen Minggang in 1988 and another one obtained by Hahn et al. in 2004. Some open problems are proposed.