On the Linear Cycle Cover Conjecture of Gy\'arf\'as and S\'ark\"ozy

TL;DR

This paper proves that every 3-uniform hypergraph's vertices can be covered by at most as many linear cycles as its largest independent set, advancing understanding of hypergraph cycle coverings.

Contribution

It establishes a weaker version of Gyárfás and Sárközy's conjecture, showing vertex coverage by linear cycles is bounded by the independence number.

Findings

Vertex set of every 3-uniform hypergraph can be covered by at most α(H) linear cycles.

Accepted a vertex and a hyperedge as a linear cycle for coverage.

Proved a weaker form of the 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 Gy\'arf\'as and S\'ark\"ozy.