A graph homomorphism from a simple graph 
to a simple graph 
is a function 
which preserves adjacency: if 
and 
are adjacent vertices
of 
,
then 
and 
are adjacent vertices of 
. Here, 
and 
are the vertex sets. The
function need not be injective
or surjective.
A proper vertex coloring with 
colors is equivalent to a graph homomorphism to the complete
graph 
.
The number of graph homomorphisms from 
to 
is denoted 
; such counts occur in the strong
Roberson conjecture.
