  At sufficiently high densities, cold dense three-flavor quark matter is in
the color-flavor locked (CFL) phase, in which all nine quarks pair in a
particularly symmetric fashion. Once the heaviness of the strange quark (mass
$m_s$) and the requirements of electric and color neutrality are taken into
account, the CFL pattern of color superconductivity requires the pairing of
quarks that would, in the absence of pairing, have Fermi momenta that differ by
of order $m_s^2/\mu$, with $\mu$ the quark number chemical potential. This
means that at sufficiently small $\mu$, the ``stress'' on the pairing is large
enough that the system can lower its energy by breaking pairs, resulting in
some unconventional color superconductor which includes gapless excitations,
spatial inhomogeneity, counter-propagating currents, or all three. In this
paper we ask whether there is some less symmetric but still conventional
pattern of pairing that can evade the stress. In other words, is there a
pattern of pairing in which, once electric and color neutrality are imposed by
suitable chemical potentials, pairing only occurs among those quarks whose
Fermi momenta would be equal in the absence of pairing? We use
graph-theoretical methods to classify 511 patterns of conventional color
superconducting pairing, and show that none of them meet this requirement. All
feel a stress, and all can be expected to become unstable to gapless modes at a
density comparable to that at which the CFL phase becomes unstable.
