AI 中文总结
该论文证明在伪随机图中，满足一定最小度条件的生成子图在全局有界着色下必含彩虹哈密顿圈，并给出计数、渗流及避免边对冲突的推广，回答了随机图上的相关问题。
AI 中文摘要
对于每个固定的 $\varepsilon\in(0,1/2)$，我们证明：若一个 $n$ 顶点 $(p,\beta)$-双混杂图 $G$ 的任意生成子图 $H$ 满足 $\delta(H)\geq(1/2+\varepsilon)pn$，则在每个全局 $\mu pn$-有界着色下，$H$ 包含一个彩虹哈密顿圈，前提是 $\beta\leq cpn$ 且 $pn\geq M$ 同时成立。同样的结论在相对条件 $°_H(v)\geq(1/2+\varepsilon)°_G(v)$（对每个顶点 $v$）下也成立，前提是 $\delta(G)\geq(1-\varepsilon/4)pn$。在任一度数条件下，这样的圈至少有 $(apn)^n$ 个。这里 $c,\mu,a,M>0$ 仅依赖于 $\varepsilon$；特别地，$pn$ 可以是足够大的常数。若 $pn\geq D\log n$，则在固定一个着色图 $H$ 后，以概率 $D\log n/(pn)$ 独立保留每条边，渐近几乎必然保持彩虹哈密顿性。对数度数要求仅用于此渗流结论。存在性定理回答了 Coulson、Keevash、Perarnau 和 Yepremyan 关于随机图的问题，并将其推广到确定性伪随机宿主图。事实上，所有三个结论在将彩虹性替换为避免指定的边对（每条边至多有 $\mu pn$ 个冲突伙伴）时仍然成立。我们在无冲突哈密顿圈上构造了一个 $O(1/(pn))$-展开的概率测度；这进而产生了枚举和渗流结果。
英文摘要
For every fixed $\varepsilon\in(0,1/2)$, we prove that every spanning subgraph $H$ of an $n$-vertex $(p,β)$-bijumbled graph satisfying $δ(H)\geq(1/2+\varepsilon)pn$ contains a rainbow Hamilton cycle under every globally $μpn$-bounded colouring, provided $β\leq cpn$ and $pn\geq M$ both hold. The same assertion holds under the relative condition $°_H(v)\geq(1/2+\varepsilon)°_G(v)$ for every vertex $v$, provided $δ(G)\geq(1-\varepsilon/4)pn$ holds. Under either degree condition, there are at least $(apn)^n$ such cycles. Here, $c,μ,a,M>0$ depend only on $\varepsilon$; in particular, $pn$ may be a sufficiently large constant. If $pn\geq D\log n$, then, upon fixing a coloured $H$, retaining each edge independently with probability $D\log n/(pn)$ preserves rainbow Hamiltonicity asymptotically almost surely. The logarithmic degree requirement is needed only for this percolation conclusion. The existence theorem answers a problem of Coulson, Keevash, Perarnau and Yepremyan for random graphs and extends it to deterministic pseudorandom hosts. In fact, all three conclusions hold with rainbowness replaced by avoidance of prescribed pairs of edges; each edge having at most $μpn$ conflicting partners. We construct an $O(1/(pn))$-spread probability measure on conflict-free Hamilton cycles; this then yields the enumeration and percolation results.
