Definition of graph homomorphisms

Suppose G = ( V, E ) and H = ( V',  E' )  are graphs. A homomorphism  from G  to H  is a mapping f  from V to V'  such that for each edge  x y  of G,  f ( x ) f ( y )   is an edge of H.  A homomorphism from G to K_n  is equivalent to an n - coloring of G.  So graph homomorphism is a generalization of coloring. A homomorphism from G  to H   is also called an  H -coloring of G .  We say G  is H -colorable if there is a homomorphism from G  to H.

Given a graph H , the H -col problem is the following decision problem.

               Instance:  a graph G.

               Question: Is G  H-colorable ?