Monochromatic Sinks in k-Arc Colored Tournaments

Abstract

Let T be a tournament on n vertices whose arcs are colored with k colors. A 3-cycle whose arcs are colored with three distinct colors is called a rainbow triangle. A rainbow triangle dominated by any nonempty set of vertices is called a dominated rainbow triangle. We prove that when $n\ge 5$, if T does not contain a dominated rainbow triangle and all 4- and 5-cycles of T are near-monochromatic, then T has a monochromatic sink. We also prove that when $n\ge 4$, if T does not contain a dominated rainbow triangle and all 4-cycles are monochromatic, then T has a monochromatic sink. A semi-cycle is a digraph C that either is a cycle or contains an arc xy such that $C-xy+yx$ is a cycle. We prove that if $n\ge 4$ and all 4-semi-cycles of T are near-monochromatic, then T has a monochromatic sink. We also show if $n\ge 5$ and all 5-semi-cycles of T are near-monochromatic, then T has a monochromatic sink.