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.
Strictly speaking, every object in every category has a universal property: an object $c \in C$ is always the universal object equipped with a map to $c$. Equivalently, universal properties usually cash out to describing the functor an object represents, and every object always represents the functor $\text{Hom}(c, -)$. In practice, often what we mean by an object having a universal property is that $\text{Hom}(c, -)$ admits some other description.
So $n$-cycles, in particular, are universal $n$-cycles: they represent the functor sending a graph to its set of $n$-cycles, where the precise definition of $n$-cycle depends on the choice of morphisms in the category of graphs.