  A hyperbolic transcendental entire function with connected Fatou set is said
to be of disjoint type. It is known that the Julia set of a disjoint-type
function of finite order is a Cantor bouquet; in particular, it is a collection
of arcs (''hairs''), each connecting a finite endpoint to infinity. We show
that the latter property is equivalent to the function being criniferous (a
necessary condition for having a Cantor bouquet Julia set). On the other hand,
we show that there is a criniferous disjoint-type entire function whose Julia
set is not a Cantor bouquet. We also provide a new characterisation of Cantor
bouquet Julia sets in terms of the existence of certain absorbing sets for the
set of escaping points, and use this to give a new intrinsic description of a
class of entire functions previously introduced by the first author. Finally,
the main known sufficient condition for Cantor bouquet Julia sets is the
so-called head-start condition of Rottenfusser et al. Under a mild geometric
assumption, we prove that this condition is also necessary.
