Coloring Problems on Bipartite Graphs of Small Diameter

We investigate a number of coloring problems restricted to bipartite graphs with bounded diameter. First, we the $k$-List Coloring, $k$-Coloring, and $k$-Precoloring Extension on diameter at most $d$, proving $\textsf{NP}$-completeness in cases, leaving open only List $3$-Coloring $3$-Precoloring when $d=3$.
 Some these results are obtained $\textsc{through}$ proof that Surjective $C_6$-Homomorphism problem is $\textsf{NP}$-complete four. Although latter result has been already proved [Vikas, 2017], present ours as an alternative simpler one. As byproduct, also get $3$-Biclique Partition $\textsf{NP}$-complete. An attempt prove this was presented [Fleischner, Mujuni, Paulusma, Szeider, 2009], but there flaw their proof, which identify discuss here.
 Finally, $3$-Fall Coloring four, for three would imply three, thus closing previously mentioned cases. This answer question posed [Kratochvíl, Tuza, Voigt, 2002].