等大小彩虹森林的近似最优填充
Nearly optimal packings of equally sized rainbow forests
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中文总结 AI 辅助
该论文证明在满足颜色类大小限制的边染色图中，可渐近最优地填充等大小的彩虹森林，解决了相关猜想在给定范围内的情形。
中文摘要 AI 辅助
边染色图中的森林若其边的颜色两两不同，则称为彩虹森林。我们证明，对于每个固定的 $0<\delta<1$，每个具有 $km$ 条边且每个颜色类大小至多 $m$ 的正常边染色简单图，当 $m\to\infty$ 时，在 $1\leq k\leq(2-\delta)m$ 范围内一致地包含至少 $(1-o(1))m$ 个两两边不相交的彩虹森林，每个森林恰好有 $k$ 条边。这确立了 Montgomery、Pokrovskiy 和 Sudakov 猜想在 $k$ 边表述下整个范围内的填充结论，并保持了原有的全局颜色界限。森林的数量是渐近最优的，且 $k$ 范围中的前导常数 $2$ 是最佳可能的。证明结合了随机星森林与二分超图的匹配定理。
英文摘要
A forest in an edge-colored graph is rainbow if its edges have pairwise distinct colors. We prove that for every $\varepsilon>0$ and all sufficiently large integers $m$, every properly edge-colored simple graph with $km$ edges, where $1\leq k\leq 2m$ and every color class has size at most $m$, contains at least $(1-\varepsilon)m$ pairwise edge-disjoint rainbow forests, each with exactly $k$ edges. The range $k\leq 2m$ is best possible: for every $k>2m$ there are such graphs containing no $k$-edge forest. Thus the conjecture of Montgomery, Pokrovskiy, and Sudakov fails beyond this range, while our theorem establishes its predicted conclusion throughout the largest possible range of $k$. The number of forests is asymptotically optimal. The proof uses an orientation dichotomy, hypergraph matching, matroid intersection, and martingale concentration.
发表机构
- School of Data Science and Information Engineering, Guizhou Minzu University（贵州民族大学数据科学与信息工程学院）
机构由 AI 辅助整理，请以论文原文为准。
