Some equitably 3-colourable cycle decompositions of complete equipartite graphs

Abstract

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 an m -cycle C in G has n i vertices coloured c i , i = 1 , 2 , … , k , and | n i - n j | ⩽ 1 for any i , j ∈ { 1 , 2 , … , k } , then C is said to be equitably k -coloured. An m -cycle decomposition C of a graph G is equitably k -colourable if the vertices of G can be coloured so that every m -cycle in C is equitably k -coloured. For m = 3, 4 and 5 we completely settle the existence question for equitably 3-colourable m -cycle decompositions of complete equipartite graphs.