Bounds on the lettericity of graphs

Lettericity measures the minimum size of an alphabet needed to represent a graph as a letter graph, where vertices are encoded by letters, and edges are determined by an underlying decoder. We prove that all graphs on~$n$ vertices have lettericity at most approximately $n-\frac{1}{2}{\log }_{2}n$ and that almost all graphs on $n$ vertices have lettericity at least $n-(2{\log }_{2}n+2{\log }_2{\log }_{2}n)$.