Partitioning edge-coloured complete graphs into monochromatic cycles and paths

A conjecture of Erd\H{o}s, Gy\'arf\'as, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertex-disjoint monochromatic cycles. So far, this conjecture has been proven only for r = 2. In this paper we show that in fact this conjecture is false for all r > 2. In contrast to this, we show that in any edge-colouring of a complete graph with three colours, it is possible to cover all the vertices with three vertex-disjoint monochromatic paths, proving a particular case of a conjecture due to Gy\'arf\'as. As an intermediate result we show that in any edge-colouring of the complete graph with the colours red and blue, it is possible to cover all the vertices with a red path, and a disjoint blue balanced complete bipartite graph.