  We study the multicritical behavior for the semimetal-insulator transitions
on graphene's honeycomb lattice using the Gross-Neveu-Yukawa effective theory
with two order parameters: the SO(3) (Heisenberg) order parameter describes the
antiferromagnetic transition, and the $\mathbb{Z}_2$ (Ising) order parameter
describes the transition to a staggered density state. Their coupling induces
multicritical behavior which determines the structure of the phase diagram
close to the multicritical point. Depending on the number of fermion flavors
$N_f$ and working in the perturbative regime in vicinity of three (spatial)
dimensions, we observe first order or continuous phase transitions at the
multicritical point. For the graphene case of $N_f=2$ and within our low order
approximation, the phase diagram displays a tetracritical structure.
