Colouring even cycle systems

An m-cycle system of order n   is a partition of the edges of the complete graph KnKn into m-cycles. An m-cycle system S is said to be weakly k-colourable if its vertices may be partitioned into k sets (called colour classes) such that no m-cycle in S has all of its vertices the same colour. The smallest value of k for which a cycle system S admits a weak k-colouring is called the chromatic number of S. We study weak colourings of even cycle systems (i.e. m-cycle systems for which m   is even), and show that for any integers r⩾2r⩾2 and k⩾2k⩾2, there is a (2r)(2r)-cycle system with chromatic number k.