Kernels by monochromatic paths and the color-class digraph

Abstract

An m-coloured digraph is a digraph whose arcs are coloured with m colors. A directed path is monochromatic when its arcs are coloured alike. A set S ⊆ V (D) is a kernel by monochromatic paths whenever the two following conditions hold: 1. For any x, y ∈ S, x 6= y, there is no monochromatic directed path between them. 2. For each z ∈ (V (D)− S) there exists a zS-monochromatic directed path In this paper it is introduced the concept of color-class digraph to prove that if D is an m-coloured strongly connected finite digraph such that: (i) Every closed directed walk has an even number of color changes, (ii) Every directed walk starting and ending with the same color has an even number of color changes, then D has a kernel by monochromatic paths. This result generalizes a classical result by Sands, Sauer and Woodrow which asserts that any 2-coloured digraph has a kernel by monochromatic paths, in case that the digraph D be a strongly connected digraph. 2000 Mathematics Subject classification: 05C20.