Do the dark circles under my child's eyes mean he has allergies?

No: Not necessarily. The dark circles under the eyes of children are usually signs of nasal congestion, which causes the blood flow to pool under the eyes. Of course, one cause of nasal congestion is allergies, but it's not the only cause. If the circles are very dark or new in onset, it's probably worth discussing with your doctor.

Yes: (possibly). Dark circles under the eyes are sometimes called "allergic shiners." they occasionally indicate the presence of allergies, but sometimes are simply seen in children with fair complexions even in the absence of allergy.