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A sequence of nonnegative integers is $k$-graphic if it is the degree sequence of a $k$-uniform hypergraph. The only known characterization of $k$-graphic sequences is due to Dewdney in 1975. As this characterization does not yield an efficient algorithm, it is a fundamental open question to determine a more practical characterization. While several necessary conditions appear in the literature, there are few conditions that imply a sequence is $k$-graphic. In light of this, we present sharp sufficient conditions for $k$-graphicality based on a sequence's length and degree sum.Kocay and Li gave a family of edge exchanges (an extension of 2-switches) that could be used to transform one realization of a 3-graphic sequence into any other realization. We extend their result to $k$-graphic sequences for all $k \geq 3$. Finally we give several applications of edge exchanges in hypergraphs, including generalizing a result of Busch et al. on packing graphic sequences.

Given graphs $G$ and $H$, $G$ is $H$-saturated if $H$ is not a subgraph of $G$, but for all $e \notin E(G)$, $H$ appears as a subgraph of $G + e$. While for every $n \ge |V(H)|$, there exists an $n$-vertex graph that is $H$-saturated, the same does not hold for induced subgraphs. That is, there exist graphs $H$ and values of $n \ge |V(H)|$, for which every $n$-vertex graph $G$ either contains $H$ as an induced subgraph, or there exists $e \notin E(G)$ such that $G + e$ does not contain $H$ as an induced subgraph. To circumvent this Martin and Smith make use of a generalized notion of "graph" when introducing the concept of induced saturation and the induced saturation number of graphs. This allows for edges that can be included or excluded when searching for an induced copy of $H$, and the induced saturation number is the minimum number of such edges that are required.In this paper, we show that the induced saturation number of many common graphs is zero. This yields graphs that are $H$-induced-saturated. That is, graphs such that no induced copy of $H$ exists, but adding or deleting any edge creates an induced copy of $H$. We introduce a new parameter for such graphs, indsat*($n;H$), which is the minimum number of edges in an $H$-induced-saturated graph. We provide bounds on indsat*($n;H$) for many graphs. In particular, we determine indsat*($n;H$) completely when $H$ is the paw graph $K_{1,3}+e$, and we determine indsat*(n;$K_{1,3}$) within an additive constant of four.

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A graphic sequence π is potentially H-graphic if there is some realization of π that contains H as a subgraph. The Erdős-Jacobson-Lehel problem asks to determine σ(H, n), the minimum even integer such that any n-term graphic sequence π with sum at least σ(H, n) is potentially H-graphic. The parameter σ(H, n) is known as the potential function of H, and can be viewed as a degree sequence variant of the classical extremal function ex(n, H). Recently, Ferrara, LeSaulnier, Moffatt and Wenger [On the sum necessary to ensure that a degree sequence is potentially H-graphic, Combinatorica 36 (2016), 687-702] determined σ(H, n) asymptotically for all H, which is analogous to the Erdős-Stone-Simonovits Theorem that determines ex(n, H) asymptotically for nonbipartite H.In this paper, we investigate a stability concept for the potential number, inspired by Simonovits' classical result on the stability of the extremal function. We first define a notion of stability for the potential number that is a natural analogue to the stability given by Simonovits. However, under this definition, many families of graphs are not σ-stable, establishing a stark contrast between the extremal and potential functions. We then give a sufficient condition for a graph H to be stable with respect to the potential function, and characterize the stability of those graphs H that contain an induced subgraph of order α(H) + 1 with exactly one edge.