  Scrutinizing the scattering of ordinary mesons in the limiting case of the
number of colour degrees of freedom $N_{\rm c}$ of quantum chromodynamics
approaching infinity, we formulate Feynman-diagram selection criteria and from
these deduce rigorous self-consistency conditions for the manifestation of a
tetraquark, a two-quark-two-antiquark bound state, as a pole in the
corresponding amplitudes. Our constraints bear rather far-reaching
consequences: In particular, all flavour-exotic tetraquarks, composed of four
(anti)quarks of disparate flavour, must come in, at least, two variants
differing in (and thus readily identifiable by) the large-$N_{\rm c}$ behaviour
of their couplings to two ordinary mesons. Quite generally, irrespective of
their flavour composition, all tetraquarks prove to be narrow. Their decay
rates behave, for large $N_{\rm c},$ like $1/N_{\rm c}^2$ and thus decrease
faster than those of ordinary mesons.
