  We investigate a lattice model of comb polymers and derive bounds on the
exponential growth rate of the number of embeddings of the comb. A comb is
composed of a backbone that is a self-avoiding walk and a set of $t$ teeth,
also modelled as mutually and self-avoiding walks, attached to the backbone at
vertices or nodes of degree 3. Each tooth of the comb has $m_a$ edges and there
are $m_b$ edges in the backbone between adjacent degree 3 vertices and between
the first and last nodes of degree 3 and the end vertices of degree 1 of the
backbone. We are interested in the exponential growth rate as $t \to \infty$
with $m_a$ and $m_b$ fixed. We prove upper bounds on this growth rate and show
that for small values of $m_a$ the growth rate is strictly less than that of
self-avoiding walks.
