Partitioning edge-coloured complete graphs into monochromatic cycles

Abstract

A conjecture of Erdos, Gyarfas, 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.