  A classic result of Asplund and Gr\"unbaum states that intersection graphs of
axis-aligned rectangles in the plane are $\chi$-bounded. This theorem can be
equivalently stated in terms of path-decompositions as follows: There exists a
function $f:\mathbb{N}\to\mathbb{N}$ such that every graph that has two
path-decompositions such that each bag of the first decomposition intersects
each bag of the second in at most $k$ vertices has chromatic number at most
$f(k)$. Recently, Dujmovi\'c, Joret, Morin, Norin, and Wood asked whether this
remains true more generally for two tree-decompositions. In this note we
provide a negative answer: There are graphs with arbitrarily large chromatic
number for which one can find two tree-decompositions such that each bag of the
first decomposition intersects each bag of the second in at most two vertices.
Furthermore, this remains true even if one of the two decompositions is
restricted to be a path-decomposition. This is shown using a construction of
triangle-free graphs with unbounded chromatic number due to Burling, which we
believe should be more widely known.
