On cycle systems with specified weak chromatic number

A weak k-colouring of an m-cycle system is a colouring of the vertices of the system with k colours in such a way that no cycle of the system has all of its vertices receive the same colour. An m-cycle system is said to be weakly k-chromatic if it has a weak k-colouring but no weak ( k − 1 ) -colouring. In this paper we show that for all k ⩾ 2 and m ⩾ 3 with ( k , m ) ≠ ( 2 , 3 ) there is a weakly k-chromatic m-cycle system of order v for all sufficiently large admissible v.