Colouring 4-cycle systems with equitably coloured blocks Original Research Article

A colouring of a 4-cycle system (V,B) is a surjective mapping φ : V→Γ. The elements of Γ are colours and, for each i∈Γ, the set Ci=φ−1(i) is a colour class. If |Γ|=m, we have an m-colouring of (V,B). For every B∈B, let φ(B)={φ(x) | x∈B}. We say that a block B is equitably coloured if either |φ(B)∩Ci|=0 or |φ(B)∩Ci|=2 for every i∈Γ. Let F(n) be the set of integers m such that there exists an m-coloured 4-cycle system of order n with every block equitably coloured. We prove that: • min F(n)=3 for every n≡1 (mod 8), n⩾17, F(9)=∅, • {m | 3⩽m⩽ n+3116}⊆F(n), n≡1 (mod 16), n⩾17, • {m | 3⩽m⩽ n+2316}⊆F(n), n≡9 (mod 16), n⩾25, • for every sufficiently large n≡1 (mod 8), there is an integer m̄ such that maxF(n)⩽m̄. Moreover we show that max F(n)=m̄ for infinite values of n.