3-uniform hypergraphs and linear cycles

Abstract

We continue the work of Gyárfás, Győri and Simonovits [Gyárfás, A., E. Győri and M. Simonovits, On 3-uniform hypergraphs without linear cycles. Journal of Combinatorics 7 (2016), 205–216], who proved that if a 3-uniform hypergraph H with n vertices has no linear cycles, then its independence number α≥2n5. The hypergraph consisting of vertex disjoint copies of complete hypergraphs K53 shows that equality can hold. They asked whether α can be improved if we exclude K53 as a subhypergraph and whether such a hypergraph is 2-colorable.We answer these questions affirmatively. Namely, we prove that if a 3-uniform linear-cycle-free hypergraph H, doesn't contain K53 as a subhypergraph, then it is 2-colorable. This result clearly implies that α≥⌈n2⌉. We show that this bound is sharp.Gyárfás, Győri 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≥10.