Let G be a graph in which each vertex has been coloured using one of k colours, say C_{1}, c_{2}, …, c_{k}. If a graph H in G has n_{i} vertices coloured c_{i} i = 1, 2, ... , k, and |n_{i} –n_{j}| __<__ 1 for any i , j Є {1, 2, . . . , k}, then H is said to be equitably k-coloured. An H-decomposition ɧ of G is equitably k-colourable if the vertices of G can be coloured so that every copy of H in ɧ is equitably k-coloured.

In Chapters 2 to 5, we consider equitably colourable decompositions. In Chapter 2, we completely settle the existence question for equitably k-colourable v-cycle decompositions of K_{v}, for 1 __<__ k __<__ v, and for any prime p we completely settle the existence question for equitably k-colourable i-perfect p-cycle decompositions of K_{p}, where 1 __<__ k __<__ p and 1 __<__ i __< __(p - 1) /2. We also completely settle the existence question for equitably 2-colourable m-cycle decompositions of K_{v} for m even and m = 5. Furthermore, we show that for all admissible v > 5, there exists at least one 5-cycle decomposition of K_{v} which cannot be equitably 2-coloured. In addition, we completely settle the existence question for equitably 2-colourable m-cycle decompositions of K_{v} - F and for equitably 3-colourable m-cycle decompositions of K_{v} and K_{v} - F, where m Є { 4, 5, 6}. We also provide upper bounds on admissible values of v for existence of equitably (m - 1)-colourable m-cycle decompositions of K_{v} and K_{v} - F. In Chapter 3, we partially generalise our results on equitably 2-colourable even-length cycle decompositions of K_{v} - F. Except for the case where v(v - 2)/2 is an odd multiple of m and v = m = 4 (mod8), we show that if the obvious necessary conditions are satisfied then there exists an equitably 2-coloured m-cycle decomposition of K_{v} - F for even m. In Chapter 4, we completely settle the existence question for equitably 2-colourable 3- and 5-cycle decompositions of K_{p}(n), and for equitably 2-colourable 4- and 6-cycle decompositions of K_{n1},_{n2}, ... ,_{np},· In addition, we completely settle the existence question for equitably 3-colourable m-cycle decompositions of K_{p}(_{n}), for m Є {3, 4, 5}. In Chapter 5, we completely settle the existence question for equitably 2- and 3-colourable 3-cube decompositions of K_{v}, K_{v} - F and K_{x},_{y}·

We also consider other types of coloured graph decompositions. Suppose that the vertices of K_{v}, have been coloured with at most two colours. Let C_{1} C_{2}, . . .C_{m} denote the colouring of the m-cycle (x_{1},x_{2}, . . . ,x_{m}) which assigns the colour C_{i} to the vertex x_{i} for i = 1, 2, . . . , m, where C_{i} Є {black, white}. We let T be the set of all possible such colourings and we let S __C__ T. If ɧ is an m-cycle system such that the colouring type of every m-cycle in ɧ is in S, and every colouring type in S is represented in ɧ, then we say that ɧ has proper colouring Type S.

In Chapter 6, we completely settle the existence question for 4-cycle decompositions with proper colouring Type S for all possible S. In Chapter 7, we completely settle the existence question for 5-cycle decompositions with proper colouring Type S for all S when |S| = 1, and for many |S| when |S| = 2. For the remaining S, where |S| = 2, we determine some necessary conditions for existence of such decompositions.