Bollobás-Meir TSP猜想在渐近意义上成立
Bollobás-Meir TSP Conjecture Holds Asymptotically
AI总结：
本文研究了Bollobás-Meir旅行商猜想，证明了渐近情况下常数c_k的上界，并修正了猜想，同时探讨了相关问题如哈密顿路径、生成树和完美匹配。
摘要：
1992年，Bollobás和Meir证明了对于每一个$k\geq 1$，存在一个常数$c_k$，使得对于任何$n$个位于$k$维单位立方体$[0, 1]^k$中的点，可以找到一个经过这些点的路线$x_1, \dots, x_n$，使得$\sum_{i = 1}^n |x_i - x_{i + 1}|^k \leq c_k$，其中$x_{n + 1} = x_1$，且$|x - y|$是$x$和$y$之间的欧几里得距离。值得注意的是，这个界限不依赖于$n$，即点的数量。他们推测最优常数是$c_k = 2 \cdot k^{k / 2}$，并证明这个下限无法更低。最近，Balogh、Clemen和Dumitrescu对$k = 3$进行了修正，证明$c_3 \geq 2^{7/2} > 2 \cdot 3^{3/2}$。对于所有$k > 2$，该猜想仍然开放，目前最好的上界是$c_k \leq 2.65^k \cdot k^{k / 2} \cdot (1 + o_k(1))$。我们显著缩小了$c_k$的上下界之间的差距，从指数级缩小到线性级。具体来说，我们证明了$c_k \leq 2\mathrm{e}(k + 1) \cdot k^{k / 2}$以及$c_k = k^{k / 2} \cdot (2 + o_k(1))$，后者在渐近意义上证实了猜想。我们还获得了关于哈密顿路径、生成树和完美匹配在单位立方体上相关问题的类似结果。我们的主要工具是一个新的球打包论证的泛化。
英文摘要：
In 1992, Bollobás and Meir showed that for every $k \geq 1$ there exists a constant $c_k$ such that, for any $n$ points in the $k$-dimensional unit cube $[0, 1]^k$, one can find a tour $x_1, \dots, x_n$ through these $n$ points with $\sum_{i = 1}^n |x_i - x_{i + 1}|^k \leq c_k$, where $x_{n + 1} = x_1$ and $|x - y|$ is the Euclidean distance between $x$ and $y$. Remarkably, this bound does not depend on $n$, the number of points. They conjectured that the optimal constant is $c_k = 2 \cdot k^{k / 2}$ and showed that it cannot be taken lower than that. This conjecture was recently revised for $k = 3$ by Balogh, Clemen and Dumitrescu, who showed that $c_3 \geq 2^{7/2} > 2 \cdot 3^{3/2}$. It remains open for all $k > 2$, with the best known upper bound $c_k \leq 2.65^k \cdot k^{k / 2} \cdot (1 + o_k(1))$. We significantly narrow the gap between lower and upper bounds on $c_k$, reducing it from exponential to linear. Specifically, we prove that $c_k \leq 2\mathrm{e}(k + 1) \cdot k^{k / 2}$ and $c_k = k^{k / 2} \cdot (2 + o_k(1))$, the latter establishing the conjecture asymptotically. We also obtain analogous results for related problems on Hamiltonian paths, spanning trees and perfect matchings in the unit cube. Our main tool is a new generalization of the ball packing argument used in earlier works.
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。
