Partitioning edge-coloured complete graphs into monochromatic cycles

A conjecture of Erdös, Gyárfás, 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 note we show that in fact this conjecture is false for all r ⩾ 3 . We also discuss some weakenings of this conjecture which may still be true.