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# Journal of Combinatorics

## Volume 7 (2016)

### Number 1

### On $3$-uniform hypergraphs without linear cycles

Pages: 205 – 216

DOI: http://dx.doi.org/10.4310/JOC.2016.v7.n1.a8

#### Authors

#### Abstract

We explore properties of $3$-uniform hypergraphs $H$ without linear cycles. It is surprising that even the simplest facts about ensuring cycles in graphs can be fairly complicated to prove for hypergraphs. Our main results are that $3$-uniform hypergraphs without linear cycles must contain a vertex of strong degree at most two and must have independent sets of size at least $\frac{2 \lvert V (H) \rvert}{5}$.

#### Keywords

cycles in hypergraphs, independent sets in hypergraphs, linear cycles

#### 2010 Mathematics Subject Classification

05B07, 05C65, 05D05