Grain amaranth is a pseudo-cereal and an ancient crop of Central and South America. Of the three species of grain amaranth, Amaranthus caudatus is mainly cultivated in the Andean region. To investigate the domestication history of A. caudatus and its relationship to the two wild relatives A. quitensis and A. hybridus, we used Genotyping by Sequencing (GBS) to genotype 119 amaranth accessions from the Andean region. We determined the genome sizes of the three species and compared phenotypic variation in two domestication-related traits, seed size and seed color. A population genetic analysis revealed very little genetic differentiation between the two wild species, suggesting they are the same species, but showed a strong differentiation between wild and domesticated amaranths despite evidence for a significant level of recent gene flow. Genome sizes and seed sizes were not significantly different between wild and domesticated amaranths, although a genetically distinct cluster of Bolivian accessions had significantly larger seeds. The analysis of seed size and seed color indicates that South American grain amaranth is an incompletely domesticated species, either because it was not strongly selected or because high levels of gene flow from its sympatric wild relatives counteract the fixation of key domestication traits. We sequenced de novo the A. caudatus genome with 150 x coverage resulting in a N50 of 150 kb. Additionally, we re-sequenced 120 individuals from all three grain species (A. caudatus, A. cruentus and A. hypochondriacus) and their potential wild ancestors (A. hybrudus and A. hybridus sp quitensis) with 10 x coverage to further trace the genetic background of the incomplete domestication. The genomic data together with transcriptomic information is being used for demographic analysis of amaranth domestication. Understanding the reason for incomplete domestication in amaranth can give additional insights into the process of crop domestication.