Ask Mr. Smarty Plants

There are many great dwarf cultivars of yaupon holly (Ilex vomitoria) that make dense plants that are strong growers and can be used for hedging or used individually in the garden. Both male and female plants are needed to have good fruit production. The red berries are very vibrant in the winter landscape.
The leaves and twigs contain caffeine, and American Indians used them to prepare a tea, which they drank in large quantities ceremonially and then vomited back up, lending the plant its species name, vomitoria. The vomiting was self-induced or because of other ingredients added; it doesn’t actually cause vomiting. Tribes from the interior traveled to the coast in large numbers each spring to partake of this tonic, and it was also a common hospitality drink among many groups. It remained popular as such among southeastern Americans into the 20th century and is still occasionally consumed today, with a flavor resembling another holly drink, the South American yerba mate, from Ilex paraguariensis.

In looking for a match to your plants, I wasn’t able to find a cultivar called ‘Gremici’ and suspect that it is either a brand new name or is spelled differently. The University of Arkansas plant database for dwarf yaupon holly list recommended cultivars as ‘Compacta,’ ‘Helleri’, ‘Condeaux’ (Bordeaux), ‘Nana’,  and ‘Schillings (Stokes Dwarf)’.
Also a possibility is the dwarf yaupon cultivar ‘Grey’s Littleleaf’ which has extra small leaves and reddish-purple new growth.  The NC State University Cooperative Extension has a good article on dwarf yaupon hollies that describe these “working class plants” that perform with first-class results.

Anyway, yaupon holly is usually quite a tough plant so we would like to suggest that perhaps there is a soil or drainage issue that is causing your plants to die. While they are quite tolerant of drought and moist soil conditions, they do prefer sandy soils and may have trouble if grown in heavy, water-logged soils.   The US Department of Agriculture Forest Service and USDA have a good factsheets on the species that will have some tips for growing this plant.