AI 中文总结
该研究针对含q≥3条边且有相邻边的固定图H，建立随机图无彩虹H着色的熵转移原理，扩展了随机Gallai着色相变，提供稠密模板熵向稀疏随机宿主转移的通用机制。
AI 中文摘要
设H为固定图，边数q=e(H)≥3且含两条相邻边，ℓ≥q为固定值。针对二项随机图在自然尺度p=n^{-1/m₂(H)}下的无彩虹H边着色，我们建立了熵转移原理：在该尺度的足够小常数倍以下，几乎所有宿主边可自由着色；在足够大常数倍以上，指数计数率完全由完全图上的确定性模板熵优化决定。精确的Hall调色板不等式在全范围q≤ℓ≤(q-1)^{q/(q-2)}内计算该速率，其中稠密侧基数为q-1，其鲁棒形式在端点下方产生计数稳定性。对任意固定ℓ，删除轮廓界确定了一阶多色行为，并刻画了(q-1)色速率在每种固定颜色数下持续的情况。这将随机Gallai着色的相变从三角形扩展到每种固定非匹配图，提供了将稠密模板熵转移到稀疏随机宿主的通用机制。
英文摘要
Let $H$ be a fixed graph with $e(H)\ge3$ that contains two adjacent edges, and let $\ell\ge e(H)$ be fixed. We establish an entropy-transference principle for rainbow-$H$-free edge-colourings of the binomial random graph at the natural scale $p=n^{-1/m_2(H)}$. Writing $R_{H,\ell}(G)$ for the number of such colourings and $λ(H,\ell)$ for the rainbow entropy--Turán density on complete graphs, we show that, with high probability, the per-edge logarithmic counting rate can be made arbitrarily close to $\log\ell$ below a sufficiently small constant multiple of this scale, and arbitrarily close to $λ(H,\ell)$ above a sufficiently large constant multiple. Thus the dense-side counting rate on a sparse random host is governed exactly by a deterministic entropy--Turán parameter on complete graphs. We further investigate this parameter, obtaining partial exact evaluations, corresponding counting-stability results, and its first-order asymptotic behaviour as the number of colours tends to infinity. This extends the random Gallai-colouring transition from triangles to every fixed non-matching graph containing at least three edges, and provides a general mechanism for transferring complete-graph template entropy to sparse random hosts.
