$3$-uniform hypergraphs and linear cycles

Abstract: Gy\'arf\'as, Gy\H{o}ri and Simonovits proved that if a $3$-uniform hypergraph with $n$ vertices has no linear cycles, then its independence number $\alpha \ge \frac{2n} {5}$. The hypergraph consisting of vertex disjoint copies of a complete hypergraph $K_5^3$ on five vertices, shows that equality can hold. They asked whether this bound can be improved if we exclude $K_5^3$ as a subhypergraph and whether such a hypergraph is $2$-colorable. In this paper we answer these questions affirmatively. Namely, we prove that if a $3$-uniform linear-cycle-free hypergraph doesn't contain $K_5^3$ as a subhypergraph, then it is $2$-colorable. This result clearly implies that its independence number $\alpha \ge \lceil \frac{n}{2} \rceil$. We show that this bound is sharp. Gy\'arf\'as, Gy\H{o}ri and Simonovits also proved that a linear-cycle-free $3$-uniform hypergraph contains a vertex of strong degree at most 2. In this context, we show that a linear-cycle-free $3$-uniform hypergraph has a vertex of degree at most $n-2$ when $n \ge 10$.