For l(G) [greater than or equal to] 3, Li and Zhang  showed that MCYP is NP-complete, even if the input graph G is diamond-free.
1 MCYP is NP-complete when restricted to triangle-free graphs.
MCYP on triangle-free graphs is clearly in NP: a nondeterministic algorithm needs only guess a set of cycles of the input graph, and check in polynomial time whether the cycles in the set are vertex-disjoint multicolored cycles that cover all the vertices of the graph, and whether the number of cycles in the set is no larger than a given positive number.
Our proof of the NP-completeness of MCYP is based on a reduction from the Minimum Set Cover problem.
3, we have that MCYP with restriction to triangle-free graphs is NP-complete.