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$χ$-binding functions for squares of bipartite graphs and its subclasses (2312.08759v1)

Published 14 Dec 2023 in cs.DM and math.CO

Abstract: 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 $G2$, 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 $\chi\left(B2\right)$ is $\Omega\left(\frac{\left(\omega\left(B2\right)\right)2 }{\log \omega\left(B2\right)}\right)$. 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 $G$, $\chi\left(G2\right) \leq \frac{3 \omega\left(G2\right)}{2}$. 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 $G2[A]$ and $G2[B]$ satisfies the property $P$ then $G2$ 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.

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