On monochromatic paths and monochromatic cycles in edge coloured tournaments Original Research Article

We call the tournament T an m-coloured tournament if the arcs of T are coloured with m colours. A directed cycle is called a quasi-monochromatic cycle if with at most one exception all of its arcs are coloured alike. In this paper I obtain some sufficient conditions for an m-coloured tournament T to have a vertex v such that for every other vertex x of T there is a monochromatic directed path from x to v. In particular I prove: If T is an m-coloured tournament such that every directed cycle of length at most 4 is a quasi-monochromatic cycle 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.