Given a graph G, we say a k-uniform hypergraph H on the same vertex set contains a Berge-G if there exists an injection φ : E(G) → E(H) such that e ⊆ φ(e) for each edge e ∈ E(G). A hypergraph H is Berge-G-saturated if H does not contain a Berge-G, but adding any edge to H creates a Berge-G. The saturation number for Berge-G, denoted sat k (n, Berge-G) is the least number of edges in a k-uniform hypergraph that is Berge-G-saturated. We determine exactly the value of the saturation numbers for Berge stars. As a tool for our main result, we also prove the existence of nearly-regular k-uniform hypergraphs, or k-uniform hypergraphs in which every vertex has degree r or r − 1 for some r ∈ Z, and less than k vertices have degree r − 1.
A permutation in a digraph $G=(V, E)$ is a bijection $f:V \rightarrow V$ such that for all $v \in V$ we either have that $f$ fixes $v$ or $(v, f(v)) \in E$. A derangement in $G$ is a permutation that does not fix any vertex. Bucic, Devlin, Hendon, Horne and Lund proved that in any digraph, the ratio of derangements to permutations is at most $1/2$. Answering a question posed by Bucic, Devlin, Hendon, Horne and Lund, we show that the set of possible ratios of derangements to permutations in digraphs is dense in the interval $[0, 1/2]$.
A derangement in G is a permutation that does not fix any vertex. In [1] it is proved that in any digraph, the ratio of derangements to permutations is at most 1/2. Answering a question posed in [1], we show that the set of possible ratios of derangements to permutations in digraphs is dense in the interval [0, 1/2].
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