发表机构
National Institute of Technology Calicut（印度国立卡尔库特技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文改进了超图格雷厄姆-波拉克定理的界限，将c_r的最小值从113降低到85，并提高了f_r(n)的界限。
AI 中文摘要
对于固定的r，令f_r(n)表示将n个顶点的完全r-均匀超图的边集分解为完全r-部r-均匀超图所需的最小数目。格雷厄姆-波拉克定理指出f_2(n)=n-1。已知f_r(n) ≤ (1+o(1))C(n,⌊r/2⌋)，随后改进为f_r(n) ≤ [r/2(14/15)^(r/4)+o(1)]C(n,⌊r/2⌋)。令c_r为lim_{n→∞}f_r(n)/C(n,⌊r/2⌋)。已知对于每个偶数r≥4，c_r<1，而对于奇数r，已知满足c_r<1的最小值为113。在本文中，我们将此值降低到85，并且还提供了f_r(n)已知界限的常数因子改进。
英文摘要
For a fixed $r$, let $f_r(n)$ denote the minimum number of complete $r$-partite $r$-uniform hypergraphs whose edge sets partition the complete $r$-uniform hypergraph on $n$ vertices. The Graham-Pollak theorem asserts that $f_2(n)=n-1$. Let $c_r$ be the smallest constant such that $f_r(n)\le c_r(1+o(1))\binom{n}{\lfloor r/2\rfloor}$. We give two constructions that improve upper bounds for $f_r(n)$. The first partitions $E(K_4)\times E(K_{17})$ into $44$ products of edge sets of complete bipartite graphs and yields $c_4\le11/12$, improving the previous bound of $c_4\le14/15$. The second gives an asymptotically improved exact cover for odd $r$. Combining these constructions, we prove that $c_r<1$ for every odd integer $r\ge65$, which improves upon the previous computer-assisted result of $113$. We further obtain the improved asymptotic bound $c_r\le \frac{r}{15}\left(\frac{11}{12}\right)^{r/4}+o(1)$ for general $r$. These results narrow the gap towards resolving the major open problem of determining whether $c_5<1$.
Comments15 pages
