T - preserving homomorphisms of oriented graphs ∗

A homomorphism of an oriented graph G = (V,A) to an oriented graph G = (V , A) is a mapping φ from V to V ′ such that φ(u)φ(v) is an arc in G whenever uv is an arc in G. A homomorphism of G to G is said to be T -preserving for some oriented graph T if for every connected subgraph H of G isomorphic to a subgraph of T , H is isomorphic to its homomorphic image in G. The T -preserving oriented chromatic number ~ χT (G) of an oriented graph G is the minimum number of vertices in an oriented graph G such that there exists a T -preserving homomorphism of G to G. This paper discusses the existence of T -preserving homomorphisms of oriented graphs. We observe that only families of graphs with bounded degree can have bounded T -preserving oriented chromatic number when T has both in-degree and out-degree at least two. We then provide some sufficient conditions for families of oriented graphs for having bounded T -preserving oriented chromatic number when T is a directed path or