A Note on the Linear Cycle Cover Conjecture of Gyárfás and Sárközy

  • Beka Ergemlidze
  • Ervin Győri
  • Abhishek Methuku
Keywords: Loose cycles, Covering, Independence number


A linear cycle in a $3$-uniform hypergraph $H$ is a cyclic sequence of hyperedges such that any two consecutive hyperedges intersect in exactly one element and non-consecutive hyperedges are disjoint. Let $\alpha(H)$ denote the size of a largest independent set of $H$.

We show that the vertex set of every $3$-uniform hypergraph $H$ can be covered by at most $\alpha(H)$ 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.

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