AI 中文总结
研究均匀着色稀疏双混合图中彩虹结构的涌现，证明在亚线性调色板盈余下完美匹配等结构存在，并推广到精确调色板情形，利用耦合与传播测度方法。
AI 中文摘要
我们研究了在独立且均匀着色的稀疏 $(p,\eta)$-双混合图中彩虹生成结构的涌现。如果着色调色板支持一个在顶点数量上呈亚线性的盈余，那么渐近几乎必然地，我们能在完美匹配、指定有界度生成树和哈密顿圈上获得彩虹嵌入，同时渐近地保持 $p$ 和 $\eta$ 处于未着色稀疏双混合图中这些结构涌现所需的最佳已知阈值条件。事实上，调色板盈余的大小由 $p$ 的一个具体递减函数上界限制。这些结果通过 McDiarmid 型耦合论证证明。接着，我们证明宿主图的混合参数轻微增加允许精确调色板结果。即，我们证明渐近几乎必然地，彩虹完美匹配、团因子、指定有界度生成树和哈密顿圈在 $(p,\eta)$-双混合图中涌现，该图的边从大小与目标配置大小一致的调色板中均匀着色。我们的精确调色板结果实际上更强；它们断言上述配置在均匀着色后不仅能在原始宿主图中以彩虹方式存活，还能在其渗透子图中存活。这些结果通过传播测度证明。
英文摘要
We study rainbow spanning configurations in bijumbled graphs whose edges are coloured independently and uniformly from a prescribed palette. For $n$-vertex $(p,β)$-bijumbled graphs with minimum degree at least a fixed positive multiple of $pn$, we obtain rainbow perfect matchings and Hamilton cycles with a sufficient palette surplus of order $(\log n)/p$, assuming $pn=ω(\log n)$ and $β\le cpn$ for a sufficiently small constant $c$. For each prescribed spanning tree of fixed maximum degree $Δ\ge2$, a surplus of order $L_{n,Δ}(\log n)/p$ suffices under $β\le cpn/L_{n,Δ}$, where $L_{n,Δ}=Δ^{5\sqrt{\log n}}$. The palette surplus is sublinear under these hypotheses. These results use a McDiarmid-type coupling and retain the discrepancy scales of the relevant deterministic embedding theorems. We also prove exact-palette results, using precisely as many colours as the number of edges of the target configuration. After independent edge percolation at rate $ρ$, it is shown that a rainbow perfect matching or Hamilton cycle exists asymptotically almost surely when $ρpn \ge C(\log n)^2$ and $β\leγpn$ for appropriate constants $C$ and $γ$. We obtain corresponding results for each prescribed bounded-degree spanning tree and for clique factors under appropriate stronger hypotheses. The exact-palette proofs construct spread measures from uncoloured containment estimates and apply the rainbow threshold theorem of Han and Yuan.
