Complements of nearly perfect graphs

A class of graphs closed under taking induced subgraphs is $χ$-bounded if there exists a function $f$ such that for all graphs $G$ in the class, $χ(G) \leq f(ω(G))$. We consider the following question initially studied in [A. Gy{á}rf{á}s, Problems from the world surrounding perfect graphs, {\em Zastowania Matematyki Applicationes Mathematicae}, 19:413--441, 1987]. For a $χ$-bounded class $\cal C$, is the class $\bar{C}$ $χ$-bounded (where $\bar{\cal C}$ is the class of graphs formed by the complements of graphs from $\cal C$)? We show that if $\cal C$ is $χ$-bounded by the constant function $f(x)=3$, then $\bar{\cal C}$ is $χ$-bounded by $g(x)=\lfloor\frac{8}{5}x\rfloor$ and this is best possible. We show that for every constant $c>0$, if $\cal C$ is $χ$-bounded by a function $f$ such that $f(x)=x$ for $x \geq c$, then $\bar{\cal C}$ is $χ$-bounded. For every $j$, we construct a class of graphs $χ$-bounded by $f(x)=x+x/\log^j(x)$ whose complement is not $χ$-bounded.

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