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Minimum pulse broadening of multimode graded-index fibers is investigated theoretically. Exact solutions of the rms pulse-width σ for a fiber with a power-law index profile is obtained by using the integral expression of group delay time based the multilayer approximation. It is shown that the minimum value of σ and the optimum α for the power-law index profile calculated by the multi-layer approximation in the steady-state modal power distribution are nearly equal to the well-known WKB solution. Next, \sigma_{\min} and the corresponding α opt are evaluated for certain index imperfections such as a central index dip and sinusoidal index ripple along a radial direction. The results offer the possibility to avoid the bandwidth degradation due to an inevitably generated index tipple in an MCVD graded-index fiber.