  Performing a leading-order renormalization group analysis, here we compute
the effects of generic local or short-range electronic interactions in
monolayer and Bernal bilayer graphene. Respectively in these two systems chiral
quasiparticles display linear and biquadratic band touching, leading to
linearly vanishing and constant DOS. Consequently, the former system remains
stable for weak enough local interactions, and supports a variety of ordered
phases only beyond a critical strength of interactions. By contrast, ordered
phases can nucleate for sufficiently weak interactions in bilayer graphene. By
tuning the strength of all symmetry allowed local interactions, we construct
various cuts of the phase diagram at zero and finite temperature and chemical
doping. Typically, at zero doping insulating phases (such as
charge-density-wave, antiferromagnet, quantum anomalous and spin Hall
insulators) prevail at the lowest temperature, while gapless nematic or smectic
liquids stabilize at higher temperatures. On the other hand, at finite doping
the lowest temperature ordered phase is occupied by a superconductor. Besides
anchoring such an organizing principle among the candidate ordered phases, we
also establish a selection rule between them and the interaction channel
responsible for the breakdown of linear or biquadrtic chiral nodal Fermi
liquid. In addition, we also demonstrate the role of the normal state band
structure in selecting the pattern of symmetry breaking from a soup of
preselected incipient competing orders. As a direct consequence of the
selection rule, while an antiferromagnetic phase develops in undoped monolayer
and bilayer graphene, the linear (biquadratic) band dispersion favors
condensation of a spin-singlet nematic (translational symmetry breaking
Kekul\'e) superconductor in doped monolayer (bilayer) graphene, when the on
site Hubbard repulsion dominates in these systems.
