Monochromatic paths and at most 2-coloured arc sets in edge-coloured tournaments

Abstract

We call the tournament T an m-coloured tournament if the arcs of T are coloured with m-colours. If v is a vertex of an m- coloured tournament T, we denote by xi(v) the set of colours assigned to the arcs with v as an endpoint. In this paper is proved that if T is an m- coloured tournament with |xi(v)| <= 2 for each vertex v of T, and T satisfies at least one of the two following properties ( 1) m not equal 3 or ( 2) m = 3 and T contains no C-3 ( the directed cycle of length 3 whose arcs are coloured with three distinct colours). Then there is a vertex v of T such that for every other vertex x of T, there is a monochromatic directed path from x to v.