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. arXiv:0808.3610v2 [math.CO]Abstract 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 (i.e. if (u, v) be the set of all vertices at out-distance 1 from v and let N 2 (v) be the set of all vertices at out-distance 2. We provide sufficient conditions under which there must exist some v ∈ V (D) such that |N 1 (v)| ≤ |N 2 (v)|, as well as examine properties of a minimal graph which does not have such a vertex. We show that if one such graph exists, then there exist infinitely many strongly-connected graphs having no such vertex. arXiv:0808.0946v3 [math.CO]