  We use the coupled cluster method in high orders of approximation to make a
comprehensive study of the ground-state (GS) phase diagram of the spin-1/2
$J_{1}$--$J_{2}$--$J_{3}$ model on a two-dimensional honeycomb lattice with
antiferromagnetic (AFM) interactions up to third-nearest neighbors. Results are
presented for the GS energy and the average local on-site magnetization. With
the nearest-neighbor coupling strength $J_{1} \equiv 1$ we find four
magnetically ordered phases in the parameter window $J_{2},J_{3} \in [0,1]$,
namely the N\'{e}el (N), striped (S), and anti-N\'{e}el (aN) collinear AFM
phases, plus a spiral phase. The aN phase appears as a stable GS phase in the
classical version of the model only for values $J_{3}<0$. Each of these four
ordered phases shares a boundary with a disordered quantum paramagnetic (QP)
phase, and at several widely separated points on the phase boundaries the QP
phase has an infinite susceptibility to plaquette valence-bond crystalline
order. We identify all of the phase boundaries with good precision in the
parameter window studied, and we find three tricritical quantum critical points
therein at: (a) $(J_{2}^{c_1},J_{3}^{c_1})=(0.51 \pm 0.01,0.69 \pm 0.01)$
between the N, S, and QP phases; (b) $(J_{2}^{c_2},J_{3}^{c_2})=(0.65 \pm
0.02,0.55 \pm 0.01)$ between the S, spiral, and QP phases; and (c)
$(J_{2}^{c_3},J_{3}^{c_3})=(0.69 \pm 0.01,0.12 \pm 0.01)$ between the spiral,
aN, and QP phases.
