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arXiv 1907.02004math.CO
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平衡$k$ -部图中的哈密顿圈

On Hamiltonian cycles in balanced $k$-partite graphs

Louis DeBiasio, Nicholas Spanier

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14 pages, 2 figures. Minor updates
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摘要：

对于所有整数$k$和$k\geq 2$，如果$G$是$n\geq 3$顶点上的平衡$k$ -部图，最小度至少为\[ \left\lceil\frac{n}{2}\right\rceil+\left\lfloor\frac{n+2}{2\lceil\frac{k+1}{2}\rceil}\right\rfloor-\frac{n}{k}=\begin{cases} \lceil\frac{n}{2}\rceil+\lfloor\frac{n+2}{k+1}\rfloor-\frac{n}{k} & : k \text{ odd }\\ \frac{n}{2}+\lfloor\frac{n+2}{k+2}\rfloor-\frac{n}{k} & : k \text{ even } \end{cases}, \]，则$G$有哈密尔顿圈，除非$k=2$和4整除$n$，或$k=\frac{n}{2}$和4整除$n$ .在$k=2$和4整除$n$，或$k=\frac{n}{2}$和4整除$n$的情况下，我们可以刻画没有哈密尔顿圈的图，并看到$\left\lceil\frac{n}{2}\right\rceil+\left\lfloor\frac{n+2}{2\lceil\frac{k+1}{2}\rceil}\right\rfloor-\frac{n}{k}+1$就足够了。对于所有$k\geq 2$和$n\geq 3$可被$k$整除，该结果是严格的。

英文摘要：

For all integers $k$ with $k\geq 2$, if $G$ is a balanced $k$-partite graph on $n\geq 3$ vertices with minimum degree at least \[ \left\lceil\frac{n}{2}\right\rceil+\left\lfloor\frac{n+2}{2\lceil\frac{k+1}{2}\rceil}\right\rfloor-\frac{n}{k}=\begin{cases} \lceil\frac{n}{2}\rceil+\lfloor\frac{n+2}{k+1}\rfloor-\frac{n}{k} & : k \text{ odd }\\ \frac{n}{2}+\lfloor\frac{n+2}{k+2}\rfloor-\frac{n}{k} & : k \text{ even } \end{cases}, \] then $G$ has a Hamiltonian cycle unless $k=2$ and 4 divides $n$, or $k=\frac{n}{2}$ and 4 divides $n$. In the case where $k=2$ and 4 divides $n$, or $k=\frac{n}{2}$ and 4 divides $n$, we can characterize the graphs which do not have a Hamiltonian cycle and see that $\left\lceil\frac{n}{2}\right\rceil+\left\lfloor\frac{n+2}{2\lceil\frac{k+1}{2}\rceil}\right\rfloor-\frac{n}{k}+1$ suffices. This result is tight for all $k\geq 2$ and $n\geq 3$ divisible by $k$.

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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