  Testing if a given graph $G$ contains the $k$-vertex path $P_k$ as a minor or
as an induced minor is trivial for every fixed integer $k\geq 1$. However, the
situation changes for the problem of checking if a graph can be modified into
$P_k$ by using only edge contractions. In this case the problem is known to be
NP-complete even if $k=4$. This led to an intensive investigation for testing
contractibility on restricted graph classes. We focus on bipartite graphs.
Heggernes, van 't Hof, L\'{e}v\^{e}que and Paul proved that the problem stays
NP-complete for bipartite graphs if $k=6$. We strengthen their result from
$k=6$ to $k=5$. We also show that the problem of contracting a bipartite graph
to the $6$-vertex cycle $C_6$ is NP-complete. The cyclicity of a graph is the
length of the longest cycle the graph can be contracted to. As a consequence of
our second result, determining the cyclicity of a bipartite graph is NP-hard.
