Journal of Combinatorial Theory Series B ( IF 1.2 ) Pub Date : 2023-07-04 , DOI: 10.1016/j.jctb.2023.06.002 Tao Jiang , Jie Ma , Liana Yepremyan
A well-known result of Verstraëte [23] shows that for each integer 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 and , there exist constants and , such that every n-vertex linear r-uniform hypergraph G with average degree contains linear cycles of k consecutive even lengths, the shortest of which is of length at most . In particular, as an immediate corollary, we retrieve the current best known upper bound on the linear Turán number of with improved coefficients.
Furthermore, we show that for any fixed integers and , there exist constants and such that every n-vertex linear r-uniform hypergraph with average degree , contains linear cycles of k consecutive lengths, the shortest of which has length at most . In both cases for given average degree d, the length of the shortest cycles cannot be improved up to the constant factors .
中文翻译:
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连续长度的线性循环
Verstraëte [23]的一个众所周知的结果表明,对于每个整数每个平均度数至少为 8 k的图G都包含k 个连续偶数长度的循环,其中最短的循环的长度最多为G半径的两倍。我们建立了 Verstraëte 在线性r均匀超图中的线性循环结果的两个扩展。
我们证明对于任何固定整数和, 存在常数和,使得每个n顶点线性r均匀超图G具有平均度包含k 个连续偶数长度的线性循环,其中最短的长度最多为。特别是,作为直接推论,我们检索了当前最著名的线性图兰数上限具有改进的系数。
此外,我们证明对于任何固定整数和, 存在常数和使得每个n顶点线性r均匀超图具有平均度,包含k 个连续长度的线性循环,其中最短的长度最多。在这两种情况下,对于给定的平均度d,最短周期的长度不能提高到常数因子。