  Lattice-based cryptography is not only for thwarting future quantum
computers, and is also the basis of Fully Homomorphic Encryption. Motivated
from the advantage of graph homomorphisms we combine graph homomorphisms with
graph total colorings together for designing new types of graph homomorphisms:
totally-colored graph homomorphisms, graphic-lattice homomorphisms from sets to
sets, every-zero graphic group homomorphisms from sets to sets. Our
graph-homomorphism lattices are made up by graph homomorphisms. These new
homomorphisms induce some problems of graph theory, for example, Number String
Decomposition and Graph Homomorphism Problem.
