  We generalize the original majority-vote (MV) model from two states to
arbitrary $p$ states and study the order-disorder phase transitions in such a
$p$-state MV model on complex networks. By extensive Monte Carlo simulations
and a mean-field theory, we show that for $p\geq3$ the order of phase
transition is essentially different from a continuous second-order phase
transition in the original two-state MV model. Instead, for $p\geq3$ the model
displays a discontinuous first-order phase transition, which is manifested by
the appearance of the hysteresis phenomenon near the phase transition. Within
the hysteresis loop, the ordered phase and disordered phase are coexisting and
rare flips between the two phases can be observed due to the finite-size
fluctuation. Moreover, we investigate the type of phase transition under a
slightly modified dynamics [Melo \emph{et al.} J. Stat. Mech. P11032 (2010)].
We find that the order of phase transition in the three-state MV model depends
on the degree heterogeneity of networks. For $p\geq4$, both dynamics produce
the first-order phase transitions.
