完成Mahmoodian-Mirzakhani猜想的边界情形及完全三部图的117种新的5-圈分解
Completing the Boundary Case of the Mahmoodian-Mirzakhani Conjecture and 117 New Computational 5-Cycle Decompositions of Complete Tripartite Graphs
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中文总结 AI 辅助
本文证明了Mahmoodian-Mirzakhani猜想的边界情形，构造了117个完全三部图的5-圈分解，解决了大量此前未解决的相关三元组的分解问题。
中文摘要 AI 辅助
设$K_{r,s,t}$（其中$r\le s\le t$）表示分划集大小为$r,s,t$的完全三部图。Mahmoodian和Mirzakhani给出了$K_{r,s,t}$可分解为5-圈的三个必要条件，并猜想这些条件是充分的，其中一个条件为$t\le 4rs/(r+s)$。我们针对极值边界$t = 4rs/(r+s)$上的所有奇数三元组证明了该猜想，证明是构造性的：在将任意奇数边界三元组约化为$(r,s,t)=(hga,hgb,ab)$（$a+b=4g$）后，我们给出$K_{ga,gb,ab}$的显式循环分解，并利用Mahmoodian和Mirzakhani的缩放定理引入公因子$h$。结合先前已知的全偶数结果，这解决了所有满足边界条件等号的三元组的猜想。我们还报告了117个满足必要条件的奇数三元组的计算机生成显式$C_5$-分解，其中116个为严格内部情形。据我们所知，这117个情形此前均未解决：未报告其中任何一个的分解，且这117个三元组均未被早期存在性结果、构造或其递归推论覆盖。此外，这117个证明连同边界构造解决了所有边数少于4400且满足必要条件的此前未解决三元组。每个计算都以机器可读的圈列表证明形式提供，可通过简短的Python验证器独立检查；我们还给出了$K_{9,19,23}$的完整人类可读边标签矩阵证明。
英文摘要
Let $K_{r,s,t}$, with $r\le s\le t$, denote the complete tripartite graph whose partite sets have sizes $r,s,t$. Mahmoodian and Mirzakhani gave three necessary conditions for $K_{r,s,t}$ to admit a decomposition into 5-cycles and conjectured that these conditions are sufficient. One of the conditions is $t\le 4rs/(r+s)$. We prove the conjecture for every odd triple on the extremal boundary $t = 4rs/(r+s)$. The proof is constructive. After reducing an arbitrary odd boundary triple to $(r,s,t)=(hga,hgb,hab)$, $a+b=4g$, we give an explicit cyclic decomposition of $K_{ga,gb,ab}$ and use the Mahmoodian and Mirzakhani scaling theorem to supply the common factor $h$. Together with the previously known all-even result, this settles the conjecture for every triple satisfying the boundary condition with equality. We also report explicit computer-generated $C_5$-decompositions for 117 odd triples satisfying the necessary conditions, 116 of which are strict-interior cases. To the best of our knowledge, all 117 cases were previously unresolved: no decomposition for any of them had been reported, and none of the 117 triples is covered by earlier existence results, constructions, or their recursive consequences. Moreover, these 117 certificates together with the boundary construction settle every previously unresolved triple satisfying the necessary conditions with fewer than $4400$ edges. Each computation is supplied as a machine-readable cycle-list certificate and can be checked independently by a short Python verifier. We also give a complete human-readable edge-label-matrix certificate for $K_{9,19,23}$.
