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. Appl. Math. 2012 .  As a byproduct, we get that deciding whether V(G) can be partitioned into 3 subsets each inducing a complete bipartite graph is NP-complete.
After 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 [ . As a byproduct, we get that deciding whether V(G) can be partitioned into 3 subsets each inducing a complete bipartite graph is NP-complete.