Can cycles – and/or other "interesting" families of graphs like paths, trees, hypercubes, etc. – be characterized by a universal property in the category of graphs?
I admit that there is not a single one category of graphs – it depends on what a graph and what graph homomorphisms are supposed to be. So the question should be:
Are there categories of graphs in which cycles (paths, trees, hypercubes, etc.) can be characterized by a universal property?
Added: I just found out, that I asked a very similar question some months ago: What is special about simplices, circles, paths and cubes?. Sorry for the duplicate.