Problems and conjectures on parameterized H-coloring

Recall that given two directed or undirected graphs G = (V (G), E(G)) and G = (V (G), E(G)) an homomorphism of G into G is a map ha θ : V (G) → V (G) with the property that {v,w} ∈ E(G) ⇒ {θ(v), θ(w)} ∈ E(G). The H-coloring problem is the problem of checking whether, for fixed H, there exists an homomorphism (H-coloring) from an input graph G to H. It is known that if H is bipartite or it has a loop the H-coloring problem can be trivially solved in polynomial time, but in the case that H is loop less and not-bipart ite the problem is known to be NP-complete [HN90]. An interesting generalization of the H-coloring problem is the list H-coloring problem where each vertex of G carries a list of the vertices of H where it is allowed to be mapped [FH98, FHH99]. In [DST01] we consider the following parameterized version of the H-coloring problem: Set up a weighting K = {kj |j ∈ C} of C ⊆ V (G) with non negative integers , we say that an input graph G has a (H,C,K)-coloring if there exists an Hhomomorphism χ : V (G) → V (H) such that ∀v ∈ C, |χ(v)| = kv . Denote (H,C,K) a partial weighted assignment. If we additionally assign to each vertex of G a list permissible images when we have a more general version of the (H,C,K)-coloring problem that we call list (H,C,K)-coloring. We can consider the integers in K to be small fixed constants, which constitutes a parameterization of the above problems problems. A weighted assignment (H,C,K) is a weighted extension of a graph F if H − C = F . In [DST01] we prove that if F is a graph where the F -coloring is NPcomplete then for any weighted extension (H,C,K) of F the (H,C,K)coloring is also NP-complete. On the other hand, if F -coloring is in P then there exist a weighted extension (H,C,K) of F so that the (H,C,K)-coloring is in P. In the same reference we also prove that if F is a graph where the list F -coloring is in P then for any weighted extension (H,C,K) of F , the