Linear cycles of consecutive lengths.

A well-known result of Verstraëte [23] shows that for each integer k ≥ 2 every graph G with average degree at least 8 k contains cycles of k consecutive even lengths, the shortest of which is of length at most twice the radius of G. We establish two extensions of Verstraëte's result for linear cycles in linear r -uniform hypergraphs. We show that for any fixed integers r ≥ 3 and k ≥ 2 , there exist constants c 1 = c 1 (r) and c 2 = c 2 (r) , such that every n -vertex linear r -uniform hypergraph G with average degree d (G) ≥ c 1 k contains linear cycles of k consecutive even lengths, the shortest of which is of length at most 2 ⌈ log ⁡ n log ⁡ (d (G) / k) − c 2 ⌉. In particular, as an immediate corollary, we retrieve the current best known upper bound on the linear Turán number of C 2 k r with improved coefficients. Furthermore, we show that for any fixed integers r ≥ 3 and k ≥ 2 , there exist constants c 3 = c 3 (r) and c 4 = c 4 (r) such that every n -vertex linear r -uniform hypergraph with average degree d (G) ≥ c 3 k , contains linear cycles of k consecutive lengths, the shortest of which has length at most 6 ⌈ log ⁡ n log ⁡ (d (G) / k) − c 4 ⌉ + 6. In both cases for given average degree d , the length of the shortest cycles cannot be improved up to the constant factors c 2 , c 4. [ABSTRACT FROM AUTHOR]