De Bruijn Covering Codes for Rooted Hypergraphs

Hypergraphs and a functional equation of Bouwkamp and de Bruijn

We show that a 1969 result of Bouwkamp and de Bruijn on a formal power series expansion can be interpreted as the hypergraph analogue of the fact that every connected graph with n vertices has at least n-1 edges. We explain some of Bouwkamp and de Bruijn's formulas in terms of hypertrees and we use Lagrange inversion to count hypertrees by the number of vertices and the number of edges of a specified size.

The de Bruijn-Erdos Theorem for Hypergraphs

Fix integers $n \ge r \ge 2$. A clique partition of ${[n] \choose r}$ is a collection of proper subsets $A_1, A_2, \ldots, A_t \subset [n]$ such that $\bigcup_i{A_i \choose r}$ is a partition of ${[n] \choose r}$. Let $\cp(n,r)$ denote the minimum size of a clique partition of ${[n] \choose r}$. A classical theorem of de Bruijn and Erd\H os states that $\cp(n, 2) = n$. In this paper we study $\cp(n,r)$, and show in general that for each fixed $r \geq 3$, \[ \cp(n,r) \geq (1 + o(1))n^{r/2} \quad \quad \mbox{as}n \rightarrow \infty.\] We conjecture $\cp(n,r) = (1 + o(1))n^{r/2}$. This conjecture…

The de Bruijn-Erdos Theorem for hypergraphs

Fix integers $n \ge r \ge 2$. A clique partition of ${[n] \choose r}$ is a collection of proper subsets $A_1, A_2, ..., A_t \subset [n]$ such that $\bigcup_i{A_i \choose r}$ is a partition of ${[n] \choose r}$. Clique partitions are related to design theory, coding theory, projective geometry, and extremal combinatorics. Let $\cp(n,r)$ denote the minimum size of a clique partition of ${[n] \choose r}$. A classical theorem of de Bruijn and Erd\H os states that $\cp(n, 2) = n$ and also determines the extremal configurations. In this paper we study $\cp(n,r)$, and show in general that for each fi…

Upper tails for counting objects in randomly induced subhypergraphs and rooted random graphs

General upper tail estimates are given for counting edges in a random induced subhypergraph of a fixed hypergraph H, with an easy proof by estimating the moments. As an application we consider the numbers of arithmetic progressions and Schur triples in random subsets of integers. In the second part of the paper we return to the subgraph counts in random graphs and provide upper tail estimates in the rooted case.

扩充超图使其具有基于拟阵的$(f,g)$-有界$(α,β)$-限制根超树填充

Augmenting a hypergraph to have a matroid-based $(f,g)$-bounded $(α,β)$-limited packing of rooted hypertrees

本文通过改编证明扩展图中根树填充理论，刻画满足顶点根树数量及总根数量上下界的基于拟阵的填充存在性，解决超图版本及扩充问题，所提方法具潜在应用价值。

超图中带根约束的有根超森林的正则打包

Regular packing of rooted hyperforests with root constraints in hypergraphs

本文研究无向图与超图中带根约束的森林正则打包问题，刻画了k个森林打包满足顶点出现次数、各森林连通分量数及总连通分量数约束的存在性条件，并推广了Bérczi和Frank的定理。