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关于$k$ -序哈密顿图

On $k$-ordered Hamiltonian Graphs

Gabor N. Sarkozy, Stanley Selkow

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摘要：

一个阶为$n$的哈密尔顿图$G$是$k$ -序的，$2\leq k \leq n$，如果对于$G$的每个$k$不同顶点的序列$v_1, v_2, \ldots ,v_k$，存在一个哈密尔顿圈，它以这个顺序遇到$v_1, v_2, \ldots , v_k$.本文回答了Ng和Schultz提出的一个问题，在适当的限制条件下，给出了保证图是$k$ -序Hamilton图的最小度的一个精确界。更精确地说，我们证明了存在$\varepsilon, n_0> 0$，使得如果$G$是一个$n\geq n_0$阶图，其最小度至少为$\lceil \frac{n}{2} \rceil + \lfloor \frac{k}{2} \rfloor - 1$和$2\leq k \leq \eps n$，则$G$是一个$k$ -阶Hamilton图.它还表明，这个界限是尖锐的每$2\leq k \leq \lfloor \frac{n}{2} \rfloor$。

英文摘要：

A Hamiltonian graph $G$ of order $n$ is $k$-ordered, $2\leq k \leq n$, if for every sequence $v_1, v_2, \ldots ,v_k$ of $k$ distinct vertices of $G$, there exists a Hamiltonian cycle that encounters $v_1, v_2, \ldots , v_k$ in this order. In this paper, answering a question of Ng and Schultz, we give a sharp bound for the minimum degree guaranteeing that a graph is a $k$-ordered Hamiltonian graph under some mild restrictions. More precisely, we show that there are $\varepsilon, n_0> 0$ such that if $G$ is a graph of order $n\geq n_0$ with minimum degree at least $\lceil \frac{n}{2} \rceil + \lfloor \frac{k}{2} \rfloor - 1$ and $2\leq k \leq \eps n$, then $G$ is a $k$-ordered Hamiltonian graph. It is also shown that this bound is sharp for every $2\leq k \leq \lfloor \frac{n}{2} \rfloor$.

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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