Homomorphisms of 2-edge-colored graphs

In this paper, we study homomorphisms of 22-edge-colored graphs, that is graphs with edges colored with two colors. We consider various graph classes (outerplanar graphs, partial 2-trees, partial 33-trees, planar graphs) and the problem is to find, for each class, the smallest number of vertices of a 22-edge-colored graph HH such that each graph of the considered class admits a homomorphism to HH.