Kernels by Monochromatic Directed Paths in \(m\)-Colored Digraphs with Quasi-Transitive Chromatic Classes

Abstract

In this paper, we give sufficient conditions for the existence of kernels by monochromatic directed paths (m.d.p.) in digraphs with quasi-transitive colorings. Let \(D\) be an \(m\)-colored digraph. We prove that if every chromatic class of \(D\) is quasi-transitive, every cycle is quasi-transitive in the rim and \(D\) does not contain polychromatic triangles, then \(D\) has a kernel by m.d.p. The same result is valid if we preserve the first two conditions before and replace the last one by: there exists \(k \geq 4\) such that every \(\overrightarrow{C}_k\) is quasi-monochromatic and every \(\overrightarrow{C}_{k-1}\) (\(3 \leq l \leq k-1\)) is not polychromatic. Finally, we also show that if every chromatic class of \(D\) is quasi-transitive, every cycle in \(D\) induces a quasi-transitive digraph and \(D\) does not contain polychromatic \(\overrightarrow{C}_3\), then \(D\) has a kernel by m.d.p. Some corollaries are obtained for the existence of kernels by m.d.p. in \(m\)-colored tournaments.