List-Compactness of Infinite Directed Graphs

Abstract. A digraph H is homomorphically compact if the digraphs G which admit homomorphisms to H are exactly the digraphs whose finite subdigraphs all admit homomorphisms to H. In this paper we define a similar notion of compactness for list-homomorphisms. We begin by showing that it is essentially only finite digraphs that are compact with respect to list-homomorphisms. We then explore the effects of restricting the types of list-assignments which are permitted, and obtain some richer characterizations.