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The slab-thickness dependence of the photonic bandgap in air-bridge photonic-crystal slabs was theoretically investigated by the 3-D plane-wave expansion method. The photonic bandgap was found to achieve a maximum for a certain slab thickness depending on air-hole volume filling factor. A kink in the photonic bandgap width versus slab thicknesses was observed, and found to be due to a transition in the nature of the modes at the top of the bandgap. Through a systematic investigation, a linear relation between the optimized slab thickness for the maximum photonic bandgap versus air-volume filling factor was found to hold over a wide range of air-volume filling factors. The linear relation also holds for photonic-crystal slabs composed of other materials and would favor the design and application of various 2-D photonic-crystal-slab devices such as reflectors, cavities, and waveguides.