论文标题
连续长度的线性循环
Linear cycles of consecutive lengths
论文作者
论文摘要
Verstraëte\ cite {v00}的一个众所周知的结果表明，对于每个整数$ k \ geq 2 $每个图$ g $，平均度至少为$ 8K $包含$ k $的循环，甚至是连续长度，最短的是$ g $的两倍。我们在线性$ r $均匀的超图中为Verstraëte的线性循环建立了两个扩展。 我们表明，对于任何固定的整数$ r \ geq 3，k \ geq 2 $，存在常数$ c_1 = c_1 = c_1（r）$和$ c_2 = c_2 = c_2（r，k）$，使每个线性$ r $ r $ r $ r $ r $ r $ - 均匀的超g $ g $ a $ 2 \ lceil \ frac {\ log n} {\ log（d（g）/k）-c_2} \ rceil $。特别是，作为一种直接的推论，我们在$ c^r_ {2k} $的线性turán数字上检索了当前最著名的上限，并带有改进的系数。 Furthermore, we show that for any fixed integers $r\geq 3,k\geq 2$, there exist constants $c_3=c_3(r)$ and $c_4=c_4(r)$ such that every $n$-vertex linear $r$-uniform graph with average degree $d(G)\geq c_3k$, contains linear cycles of $k$ consecutive lengths, the shortest of which长度最多有$ 6 \ lceil \ frac {\ log n} {\ log（d（g）/k）-c_4} -c_4} \ rceil +6 $。保证的循环中的程度条件和最短长度都可以达到恒定因素。
A well-known result of Verstraëte \cite{V00} shows that for each integer $k\geq 2$ every graph $G$ with average degree at least $8k$ contains cycles of $k$ consecutive even lengths, the shortest of which is 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 $r\geq 3,k\geq 2$, there exist constants $c_1=c_1(r)$ and $c_2=c_2(r,k)$, such that every linear $r$-uniform hypergraph $G$ with average degree $d(G)\geq c_1 k$ contains linear cycles of $k$ consecutive even lengths, the shortest of which is at most $2\lceil \frac{ \log n}{\log (d(G)/k)-c_2}\rceil$. In particular, as an immediate corollary, we retrieve the current best known upper bound on the linear Turán number of $C^r_{2k}$ with improved coefficients. Furthermore, we show that for any fixed integers $r\geq 3,k\geq 2$, there exist constants $c_3=c_3(r)$ and $c_4=c_4(r)$ such that every $n$-vertex linear $r$-uniform graph with average degree $d(G)\geq c_3k$, contains linear cycles of $k$ consecutive lengths, the shortest of which has length at most $6\lceil \frac{\log n}{\log (d(G)/k)-c_4} \rceil +6$. Both the degree condition and the shortest length among the cycles guaranteed are best possible up to a constant factor.
