Some equitably 3-colorable cycle decompositions of Kv+IKv+I

Let GG be a graph in which each vertex has been colored using one of kk colors, say c1,c2,…,ckc1,c2,…,ck. If an mm-cycle CC in GG has nini vertices colored cici, i=1,2,…,ki=1,2,…,k, and ∣ni−nj∣≤1∣ni−nj∣≤1 for every i,j∈{1,2,…,k}i,j∈{1,2,…,k}, then CC is equitably kk-colored. An mm-cycle decomposition CC of a graph GG is equitably kk-colorable if the vertices of GG can be colored so that every mm-cycle in CC is equitably kk-colored. For m=4,5 and 6, we completely settle the existence problem for equitably 33-colorable mm-cycle decompositions of complete graphs with the edges of a 1-factor added.