AI 中文总结
该研究分析n维布尔立方体的圈分解与哈密顿圈数量的对数渐近行为，证明特定完美匹配可扩展为哈密顿圈，为相关组合结构研究提供了关键渐近结果。
AI 中文摘要
研究表明，当n趋于无穷大时，n维布尔立方体E^n的圈分解数量的对数为2^n(ln n -1 + o(1))，E^n中哈密顿圈数量的对数至少为2^{n-1}(ln n -1 + o(1))；还证明了，对于每个n≥n₀(k)，E^n中每条边属于至多k个方向的完美匹配都可扩展为一个哈密顿圈。
英文摘要
It is shown that, as $n\to\infty$, the logarithm of the number of decompositions into cycles of the $n$-dimensional Boolean cube $E^n$ is \[ 2^n(\ln n-1+o(1)), \] and the logarithm of the number of Hamiltonian cycles in $E^n$ is at least \[ 2^{n-1}(\ln n-1+o(1)). \] It is proved that, in $E^n$, every perfect matching whose edges belong to at most $k$ directions can be extended to a Hamiltonian cycle for every $n\geq n_0(k)$.
CommentsThis article was published in Russian in 2001. The results of this article have been surpassed by the following papers. 1.Feder T., Subi C., Nearly tight bounds on the number of Hamiltonian circuits... Inf. Process. Lett. 109, No. 5, 267-272 (2009). 2.Fink J., Perfect matchings extend to Hamilton cycles in hypercubes. J. Comb. Theory, Ser. B 97, No. 6, 1074-1076 (2007)
Journal refDiskretn. Anal. Issled. Oper., Ser. 1, 8:2 (2001), 52-62
