  We consider maps with tight boundaries, i.e. maps whose boundaries have
minimal length in their homotopy class, and discuss the properties of their
generating functions $T^{(g)}_{\ell_1,\ldots,\ell_n}$ for fixed genus $g$ and
prescribed boundary lengths $\ell_1,\ldots,\ell_n$, with a control on the
degrees of inner faces. We find that these series appear as coefficients in the
expansion of $\omega^{(g)}_n(z_1,\ldots,z_n)$, a fundamental quantity in the
Eynard-Orantin theory of topological recursion, thereby providing a
combinatorial interpretation of the Zhukovsky transformation used in this
context. This interpretation results from the so-called trumpet decomposition
of maps with arbitrary boundaries. In the planar bipartite case, we obtain a
fully explicit formula for $T^{(0)}_{2\ell_1,\ldots,2\ell_n}$ from the
Collet-Fusy formula. We also find recursion relations satisfied by
$T^{(g)}_{\ell_1,\ldots,\ell_n}$, which consist in adding an extra tight
boundary, keeping the genus $g$ fixed. Building on a result of Norbury and
Scott, we show that $T^{(g)}_{\ell_1,\ldots,\ell_n}$ is equal to a
parity-dependent quasi-polynomial in $\ell_1^2,\ldots,\ell_n^2$ times a simple
power of the basic generating function $R$. In passing, we provide a bijective
derivation in the case $(g,n)=(0,3)$, generalizing a recent construction of
ours to the non bipartite case.
