A universal cycle is a compact listing of a class of combinatorial objects. In this paper, we prove the existence of universal cycles of classes of labeled graphs, including simple graphs, trees, graphs with $m$ edges, graphs with loops, graphs with multiple edges (with up to $m$ duplications of each edge), directed graphs, hypergraphs, and $k$-uniform hypergraphs.

Let D be a simple digraph without loops or digons. For any v in V(D) let N_1(v) be the set of all nodes at out-distance 1 from v and let N_2(v) be the set of all nodes at out-distance 2. We provide sufficient conditions under which there must exist some v in V(D) such that |N_1(v)| is less than or equal to |N_2(v)|, as well as examine properties of a minimal graph which does not have such a node. We show that if one such graph exists, then there exist infinitely many strongly-connected graphs having no such vertex.

Let D be a simple digraph without loops or digons. For any v ∈ V (D) let N1(v) be the set of all nodes at out-distance 1 from v and let N2(v) be the set of all nodes at out-distance 2. We provide conditions under which there must exist some v ∈ V (D) such that |N1(v)| ≤ |N2(v)|, as well as examine extremal properties in a minimal graph which does not have such a node. We show that if one such graph exists, then there exist infinitely many.