Equitably 2-colourable even cycle systems

An l-cycle decomposition of a graph G is said to be equitably c-colourable if there is a c-vertex-colouring of G such that each colour is represented (approximately) an equal number of times on each cycle. In this paper, we consider the existence of equitably 2-colourable even l-cycle systems of the cocktail party graph K-v - I. After establishing that the problem of proving existence of equitably 2-colourable l-cycle decompositions of K-v - I reduces to considering l-admissible values v is an element of [l, 2l), we determine a complete existence result for equitably 2-colourable l-cycle decompositions of K-v - I in the cases that v = 0,2 (mod l), or l is a power of 2, or l is an element of{2q, 4q} for q an odd prime power, or l <= 30.