  A class $\mathcal G$ of graphs is $\chi$-bounded if there is a function $f$
such that for every graph $G\in \mathcal G$ and every induced subgraph $H$ of
$G$, $\chi(H)\le f(\omega(H))$. In addition, we say that $\mathcal G$ is
polynomially $\chi$-bounded if $f$ can be taken as a polynomial function. We
prove that for every integer $n\ge3$, there exists a polynomial $f$ such that
$\chi(G)\le f(\omega(G))$ for all graphs with no vertex-minor isomorphic to the
cycle graph $C_n$. To prove this, we show that if $\mathcal G$ is polynomially
$\chi$-bounded, then so is the closure of $\mathcal G$ under taking the
$1$-join operation.
