Some equitably 3-colorable cycle decompositions of

Let G be a graph in which each vertex has been colored using one of k colors, say c 1 , c 2 , … , c k . If an m -cycle C in G has n i vertices colored c i , i = 1 , 2 , … , k , and ∣ n i − n j ∣ ≤ 1 for every i , j ∈ { 1 , 2 , … , k } , then C is equitably k -colored. An m -cycle decomposition C of a graph G is equitably k -colorable if the vertices of G can be colored so that every m -cycle in C is equitably k -colored. For m = 4 , 5 and 6, we completely settle the existence problem for equitably 3 -colorable m -cycle decompositions of complete graphs with the edges of a 1-factor added.