Before revision:  We also prove that the Surjective C_6-Homomorphism problem is NP-complete on bipartite graphs with diameter at most 4, answering a question posed by Bodirsky, K\'ara, and Martin [ Discret.
After revision:  Some of these results are obtained through a proof that the Surjective C_6-Homomorphism problem is NP-complete on bipartite graphs with diameter at most 4, answering a question posed by Bodirsky, K\'ara, and Martin [ Discret.