Hypergraph Turán numbers of vertex disjoint cycles

Abstract

The Turán number of a $k$-uniform hypergraph $H$, denoted by $e{x}_k\left(n;H\right)$, is the maximum number of edges in any $k$-uniform hypergraph $F$ on $n$ vertices which does not contain $H$ as a subgraph. Let ${\mathcal{C}}_{\ell }^{\left(k\right)}$ denote the family of all $k$-uniform minimal cycles of length $\ell$, $\mathcal{S}({\ell }_1,\dots ,{\ell }_r)$ denote the family of hypergraphs consisting of unions of $r$ vertex disjoint minimal cycles of length ${\ell }_1,\dots ,{\ell }_r$, respectively, and ${\mathbb{C}}_{\ell }^{\left(k\right)}$ denote a $k$-uniform linear cycle of length $\ell$. We determine precisely $e{x}_k\left(n;\mathcal{S}({\ell }_1,\dots ,{\ell }_r)\right)$ and $e{x}_k\left(n;{\mathbb{C}}_{{\ell }_1}^{\left(k\right)},\dots ,{\mathbb{C}}_{{\ell }_r}^{\left(k\right)}\right)$ for sufficiently large $n$. The results extend recent results of Füredi and Jiang who determined the Turán numbers for single $k$-uniform minimal cycles and linear cycles.

9 pages. arXiv admin note: text overlap with arXiv:1302.2387 by other authors

References in corpus (1)

• Hypergraph Turan numbers of linear cycles

Cited by in corpus (2)

• Turan Problems and Shadows I: Paths and Cycles
• The Turan Number of Disjoint Copies of Paths