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.

翻译：图类 $\mathcal{G}$ 称为 $\chi$ 有界的，若存在函数 $f$ 使得对任意 $G \in \mathcal{G}$ 满足 $\chi(G) \leq f(\omega(G))$，其中 $\chi(G)$ 和 $\omega(G)$ 分别表示图 $G$ 的色数和团数。图 $G$ 的平方，记作 $G^2$，是与 $G$ 具有相同顶点集的图，其中两个顶点在 $G$ 中距离不超过 2 时相邻。本文研究二部图及其子类平方的 $\chi$ 有界性。注意到一般图的平方类允许二次 $\chi$ 有界函数，且存在二部图 $B$ 使得 $\chi\left(B^2\right)$ 为 $\Omega\left(\frac{\left(\omega\left(B^2\right)\right)^2 }{\log \omega\left(B^2\right)}\right)$。我们首先提出以下问题："二部图的哪些子类具有线性 $\chi$ 有界函数？" 我们聚焦于凸二部图类，并证明以下结果：对任意凸二部图 $G$，有 $\chi\left(G^2\right) \leq \frac{3 \omega\left(G^2\right)}{2}$。该证明同时给出了多项式时间 $3/2$ 近似算法，用于对凸二部图平方着色。随后，我们针对二部图平方引入"部可测试性质"概念：称图性质 $P$ 对二部图平方是部可测试的，若对于二部图 $G=(A,B,E)$，当诱导子图 $G^2[A]$ 和 $G^2[B]$ 满足性质 $P$ 时，$G^2$ 也满足性质 $P$。本文讨论了完美性、弦性、(反空穴)-自由性等经典图性质是否具有部可测试性。作为推论，我们证明了双凸二部图的平方是完美图。