We here pioneer a low-cost assembly strategy for 20 Heliconiini genomes to characterize the evolutionary history of the rapidly radiating genus Heliconius . A bifurcating tree provides a poor fit to the data, and we therefore explore a reticulate phylogeny for Heliconius . We probe the genomic architecture of gene flow, and develop a new method to distinguish incomplete lineage sorting from introgression. We find that most loci with non-canonical histories arose through introgression, and are strongly underrepresented in regions of low recombination and high gene density. This is expected if introgressed alleles are more likely to be purged in such regions due to tighter linkage with incompatibility loci. Finally, we identify a hitherto unrecognized inversion, and show it is a convergent structural rearrangement that captures a known color pattern switch locus within the genus. Our multi-genome assembly approach enables an improved understanding of adaptive radiation.