Mona~efte ffir Maihemaiik 1Ylh. Math. 85, 39--48 (1978) 9 by Springer-Verlag 1978 Homomorphisms of Graphs and of Their Orientations By P. Hell, Vancouver, and J. Ne~et~il, Prague With 1 Figure (Received 22 March 1976; in revisedform 22 February 1977) Abstract Consider a set of graphs and all the homomorphisms among them. Change each graph into a digraph by assigning directions to its edges. Some of the homomorphisms preserve the directions and so remain as homo- morphisms of the set of digraphs; others do not. We study the relationship between the original set of graph-homomorphisms and the resulting set of digraph-homomorphisms and prove that they are in a certain sense indepen- dent. This independence result no longer holds if we start with a proper class of graphs, or if we require that only one direction be given to each edge (unless each homomorphism is invertibte, in which ease we again prove independence). We also specialize the results to the set consisting of one graph and prove the independence of monoids (groups) of a graph and the corresponding digraph. w 1 Introduction In this section we present the necessary definitions and pose the problems to be solved in the rest of the paper. The simplest instance of the general structures we shall study is a digraph. A digraph X is a set V with a binary- relation /~; we write X : (V, R). The elements of V are called the vertices and ~he elements of R, the arcs of X; we write V : V (X) and R :/~ (X) if X is not clear from the context. If X,Y are digraphs, a homomorphism f:X->Y is a mapping f:V(X)->V(Y) for which (x,x')sR(X) implies (f(x),f(x'))st~(Y). A homomorphism Z~X is called an endomorphism of X, and a bijective endomorphism is termed an automorphism. Composition of mappings gives rise to the following homomorphism-related structures : the group of all auto- morphisms of a digraph X, denoted by AutX; the monoid of all 0026--9255/78/0085/0039/8 02.00