Summary: A class of graphs closed under taking induced subgraphs is \(\chi\)-bounded if there exists a function \(f\) such that for all graphs \(G\) in the class, \(\chi(G) \leq f(\omega(G\))). We consider the following question initially studied in [A. Gyárfás, Zastosow. Mat. 19, No. 3–4, 413–441 (1987; Zbl 0718.05041)]. For a \(\chi\)-bounded class \(\mathcal{C}\), is the class \(\overline{C}\chi\)-bounded (where \(\overline{\mathcal{C}}\) is the class of graphs formed by the complements of graphs from \(\mathcal{C}\))?
We show that if \(\mathcal{C}\) is \(\chi\)-bounded by the constant function \(f(x) = 3\), then \(\overline{\mathcal{C}}\) is \(\chi\)-bounded by \(g(x) = \left\lfloor \frac{8}{5}x \right\rfloor\) and this is best possible.
We show that for every constant \(c>0\), if \(\mathcal{C}\) is \(\chi\)-bounded by a function \(f\) such that \(f(x) = x\) for \(x \geq c\), then \(\overline{\mathcal{C}}\) is \(\chi\)-bounded. For every \(j\), we construct a class of graphs \(\chi\)-bounded by \(f(x) = x+x/\log^{j}(x)\) whose complement is not \(\chi\)-bounded.