On the Linear Cycle Cover Conjecture of Gyárfás and Sárközy

Abstract

A linear cycle in a hypergraph $H$ is a cyclic sequence of hyperedges such that two consecutive hyperedges intersect in exactly one element and two nonconsecutive hyperedges are disjoint and $\alpha (H)$ denotes the size of a largest independent set of $H$. In this note, we show that the vertex set of every $3$-uniform hypergraph $H$ can be covered by at most $\alpha (H)$ pairwise edge-disjoint linear cycles (where we accept a vertex and a hyperedge as a linear cycle), proving a weaker version of a conjecture of Gyárfás and Sárközy.