Linear cycles of consecutive lengths

Abstract

A well-known result of Verstraëte [23] shows that for each integer k≥2 every graph G with average degree at least 8k 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 c1=c1(r) and c2=c2(r), such that every n-vertex linear r-uniform hypergraph G with average degree d(G)≥c1k contains linear cycles of k consecutive even lengths, the shortest of which is of length at most 2⌈log⁡nlog⁡(d(G)/k)−c2⌉. In particular, as an immediate corollary, we retrieve the current best known upper bound on the linear Turán number of C2kr with improved coefficients.Furthermore, we show that for any fixed integers r≥3 and k≥2, there exist constants c3=c3(r) and c4=c4(r) such that every n-vertex linear r-uniform hypergraph with average degree d(G)≥c3k, contains linear cycles of k consecutive lengths, the shortest of which has length at most 6⌈log⁡nlog⁡(d(G)/k)−c4⌉+6. In both cases for given average degree d, the length of the shortest cycles cannot be improved up to the constant factors c2,c4.