  We generalize the random graph evolution process of Bohman, Frieze, and
Wormald [T. Bohman, A. Frieze, and N. C. Wormald, Random Struct. Algorithms,
25, 432 (2004)]. Potential edges, sampled uniformly at random from the complete
graph, are considered one at a time and either added to the graph or rejected
provided that the fraction of accepted edges is never smaller than a decreasing
function asymptotically approaching the value alpha = 1/2. We show that
multiple giant components appear simultaneously in a strongly discontinuous
percolation transition and remain distinct. Furthermore, tuning the value of
alpha determines the number of such components with smaller alpha leading to an
increasingly delayed and more explosive transition. The location of the
critical point and strongly discontinuous nature are not affected if only edges
which span components are sampled.
