发表机构
Illinois State University（伊利诺伊州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究超图分解的嵌入问题，给出 h=2,3 的精确存在性准则，并将 h≥4 的充分条件阈值从 n≥hm 改进至 n≥(h−1)m，同时提供连通、简单及等倍数目标因子的精确准则。
AI 中文摘要
我们研究了将 λ 重完全 h 一致超图 λK_m^h 的分解嵌入到 λK_n^h 的分解中，其中 m<n，且源和目标度数可能因颜色而异。从每个嵌入出发，可以构造另一个嵌入，该嵌入在每种目标颜色中最小化分量数，而不增加其多重性跨度。这种改变也不会增加边多重性的任何凸函数之和，且在所有颜色中同时成立。对于 h=2,3，我们给出了每个 m<n 的精确存在性准则。对于 h≥4，当 n≥(h−1)m 时，必要的可除性和度数和条件即为充分条件，这改进了先前已知的针对颜色相关度数的阈值 n≥hm。在此范围内，我们还给出了连通、简单和等倍数目标因子的精确准则。证明使用了一个整数计数系统，记录每种类型添加边的数量。每个满足这些方程的嵌入都给出了在允许的最小多重性跨度下的嵌入。
英文摘要
We study embeddings of factorizations of the $λ$-fold complete $h$-uniform hypergraph $λK_m^h$ in factorizations of $λK_n^h$, where $m<n$ and both the source and target degrees may vary by color. From every embedding one can obtain another that minimizes the number of components in every target color without increasing its multiplicity spread. This change also does not increase the sum of any convex function of the edge multiplicities, simultaneously in all colors. For $h=2,3$ we give exact existence criteria for every $m<n$. For $h\ge4$, the necessary divisibility and degree-sum conditions are sufficient once $n\ge(h-1)m$, improving the previously known threshold $n\ge hm$ for color-dependent degrees. In this range we also give exact criteria for connected, simple, and equimultiple target factors. The proofs use a system of integer counts recording the number of added edges of each type. Every system satisfying these equations gives an embedding with the minimum multiplicity spread allowed by
Comments23 pages
