(\chi\)-binding functions for squares of bipartite graphs and its subclasses

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.