Homomorphism-homogeneous L-colored graphs
  A relational structure is homomorphism-homogeneous (HH-homogeneous for short) if every homomorphism between finite induced substructures of the structure can be extended to a homomorphism over the whole domain of the structure. Similarly, a structure is monomorphism-homogeneous (MH-homogeneous for short) if every monomorphism between finite induced substructures of the structure can be extended to a homomorphism over the whole domain of the structure. In this paper we consider L-colored graphs, that is, undirected graphs without loops where sets of colors selected from L are assigned to vertices and edges. A full classification of finite MH-homogeneous L-colored graphs where L is a chain is provided, and we show that the classes MH and HH coincide. When L is a diamond, that is, a set of pairwise incomparable elements enriched with a greatest and a least element, the situation turns out to be much more involved. We show that in the general case the classes MH and HH do not coincide. 
David Hartman, Jan Hubicka and Dragan Masulovic
math.CO