  A class of graphs is $\chi$-bounded if there is a function $f$ such that
every graph $G$ in the class has chromatic number at most $f(\omega(G))$, where
$\omega(G)$ is the clique number of $G$; the class is polynomially
$\chi$-bounded if $f$ can be taken to be a polynomial. The Gy\'arf\'as-Sumner
conjecture asserts that, for every forest $H$, the class of $H$-free graphs
(graphs with no induced copy of $H$) is $\chi$-bounded. Let us say a forest $H$
is good if it satisfies the stronger property that the class of $H$-free graphs
is polynomially $\chi$-bounded.
  Very few forests are known to be good: for example, it is open for the
five-vertex path. Indeed, it is not even known that if every component of a
forest $H$ is good then $H$ is good, and in particular, it was not known that
the disjoint union of two four-vertex paths is good. Here we show the latter,
and more generally, that if $H$ is good then so is the disjoint union of $H$
and a four-vertex path. We also prove a more general result: if every component
of $H_1$ is good, and $H_2$ is any path (or broom) then the class of graphs
that are both $H_1$-free and $H_2$-free is polynomially $\chi$-bounded.

Combinatorics