  A class of graphs $\mathcal{G}$ is $\chi$-bounded if there exists a function
$f$ such that $\chi(G) \leq f(\omega(G))$ for each graph $G \in \mathcal{G}$,
where $\chi(G)$ and $\omega(G)$ are the chromatic and clique number of $G$,
respectively. The square of a graph $G$, denoted as $G^2$, is the graph with
the same vertex set as $G$ in which two vertices are adjacent when they are at
a distance at most two in $G$. In this paper, we study the $\chi$-boundedness
of squares of bipartite graphs and its subclasses. Note that the class of
squares of graphs, in general, admit a quadratic $\chi$-binding function.
Moreover there exist bipartite graphs $B$ for which $\chi\left(B^2\right)$ is
$\Omega\left(\frac{\left(\omega\left(B^2\right)\right)^2 }{\log
\omega\left(B^2\right)}\right)$. We first ask the following question: "What
sub-classes of bipartite graphs have a linear $\chi$-binding function?" We
focus on the class of convex bipartite graphs and prove the following result:
for any convex bipartite graph $G$, $\chi\left(G^2\right) \leq \frac{3
\omega\left(G^2\right)}{2}$. Our proof also yields a polynomial-time
$3/2$-approximation algorithm for coloring squares of convex bipartite graphs.
We then introduce a notion called "partite testable properties" for the squares
of bipartite graphs. We say that a graph property $P$ is partite testable for
the squares of bipartite graphs if for a bipartite graph $G=(A,B,E)$, whenever
the induced subgraphs $G^2[A]$ and $G^2[B]$ satisfies the property $P$ then
$G^2$ also satisfies the property $P$. Here, we discuss whether some of the
well-known graph properties like perfectness, chordality, (anti-hole)-freeness,
etc. are partite testable or not. As a consequence, we prove that the squares
of biconvex bipartite graphs are perfect.
