  We formulate rigorous criteria for selecting diagrams to be taken into
account in the analysis of potential polyquark (tetra or penta) poles in QCD.
The central point of these criteria is the requirement that the Feynman
diagrams for the relevant Green functions contain four-quark (or, respectively,
five-quark in the pentaquark case) intermediate states and the corresponding
cuts. It is shown that some diagrams which "visually" seem to contain the
four-quark cuts, turn out to be free of these singularities and therefore
should not be taken into account when calculating the tetraquark properties. We
then consider large-$N_c$ QCD which in many cases provide qualitatively correct
picture of hadron properties and discuss in detail the tetraquark states. For
the "direct" and the "recombination" four-point Green functions, which may
potentially contain the tetraquark poles, we formulate large-$N_c$ consistency
conditions which strongly restrict the behaviour of the tetraquark-to-ordinary
meson transition amplitudes. In the end, these conditions allow us to obtain
constraints on width of the potential tetraquark states at large $N_c$. We
report that both flavor-exotic and cryptoexotic (i.e., flavor-nonexotic)
tetraquarks, if the corresponding poles exist, have a width of order
$O(1/N_c^2)$, i.e. they should be parametrically narrower compared to the
ordinary $\bar qq$ mesons with a width of order $O(1/N_c)$. Moreover, for
flavor-exotic states, the large-$N_c$ consistency conditions require two narrow
flavor-exotic states, each of these states coupling dominantly to one specific
meson-meson channel.
