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arXiv 1612.08802math.CO
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当每个圈包含k个和弦时的泛圈性

Pancyclicity when each cycle contains k chords

Vladislav Taranchuk

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摘要：

对于整数$n \geq k \geq 2$，令$c(n,k)$是必须添加到长度为$n$的循环的弦的最小数目，使得所得图具有以下性质：对于每个$l \in \{ k , k + 1 , \dots , n \}$，存在长度为$l$的循环，其恰好包含所添加的弦的$k$。Affif Chaouche、Rutherford和Whitty介绍了函数$c(n,k)$。他们证明了对于每一个整数$k \geq 2$，$c(n , k ) \geq Ω_k ( n^{1/k} )$，他们问$n^{1/k}$是否给出了，对于$k \geq 2$，$c(n, k)$的正确数量级。我们的主要定理回答了这个问题，因为我们证明，对于每个整数$k \geq 2$，并为足够大的$n$，$c(n , k) \leq k \lceil n^{1/k} \rceil + k^2$。这个上限，连同下界的Affif Chaouche等。例如，显示$c(n,k)$的数量级为$n^{1/k}$。

英文摘要：

For integers $n \geq k \geq 2$, let $c(n,k)$ be the minimum number of chords that must be added to a cycle of length $n$ so that the resulting graph has the property that for every $l \in \{ k , k + 1 , \dots , n \}$, there is a cycle of length $l$ that contains exactly $k$ of the added chords. Affif Chaouche, Rutherford, and Whitty introduced the function $c(n,k)$. They showed that for every integer $k \geq 2$, $c(n , k ) \geq Ω_k ( n^{1/k} )$ and they asked if $n^{1/k}$ gives the correct order of magnitude of $c(n, k)$ for $k \geq 2$. Our main theorem answers this question as we prove that for every integer $k \geq 2$, and for sufficiently large $n$, $c(n , k) \leq k \lceil n^{1/k} \rceil + k^2$. This upper bound, together with the lower bound of Affif Chaouche et.\ al., shows that the order of magnitude of $c(n,k)$ is $n^{1/k}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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