随机图在不同密度区间的多重集染色
Multiset Colorings of Random Graphs Across Density Regimes
- Tehran Institute for Advanced Studies (TeIAS), Khatam University（德黑兰高等研究院，哈塔姆大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究随机图的多重集色数，证明在广泛密度区间内几乎处处有界，并给出接近完全图时的精确增长阶。
AI中文摘要：
我们证明几乎所有的图都允许将其顶点集划分为三个部分，使得任意两个相邻顶点在每个部分中的邻居数都不相同。等价地，对于 $G\sim G(n,1/2)$，$\u03c7_m(G)\le3$ 以高概率成立，改进了先前已知的界五。这里 $\u03c7_m(G)$ 表示图 $G$ 的多重集色数，即顶点划分中使得邻居计数向量区分每对相邻顶点的最少部分数。事实上，对于每个固定的 $0.185<p<0.509$，三部分界都成立。更一般地，对于每个固定的 $p\in(0,1)$，$G\sim G(n,p)$ 满足 $\u03c7_m(G)\le4$ 以高概率成立。这些结果是通过将精心选择的初始划分的未解析边转换为布尔立方体的超平面，并应用 Linial--Radhakrishnan 本质覆盖理论获得的。我们还确定了当图多项式接近完全图时 $\u03c7_m(G)$ 的增长情况。对于每个固定的 $\beta\in(0,1)$ 和 $G\sim G\\!\left(n,1-n^{-(1-\beta)}\right)$，以高概率有 $\frac{2}{\beta}\le \u03c7_m(G)\le \left\lfloor\frac{2}{\beta}\right\rfloor+5$。因此 $\u03c7_m(G)=2/\beta+O(1)$。下界是谱界的，而上界则来自多项式反集中和 Lov'asz 局部引理。
英文摘要：
We show that almost every graph admits a partition of its vertex set into three parts such that no two adjacent vertices have the same number of neighbors in each of the three parts. Equivalently, for $G\sim G(n,1/2)$, $χ_m(G)\le3$ with high probability, improving the previously known bound of five. Here $χ_m(G)$ denotes the multiset chromatic number of $G$, the minimum number of parts in a vertex partition whose neighbor-count vectors distinguish every pair of adjacent vertices. In fact, the three-part bound holds for every fixed $0.185<p<0.509$. More generally, for every fixed $p\in(0,1)$, $G\sim G(n,p)$ satisfies $χ_m(G)\le4$ with high probability. These results are obtained by converting the unresolved edges of a carefully chosen initial partition into hyperplanes of a Boolean cube and applying the Linial--Radhakrishnan theory of essential covers. We also determine how $χ_m(G)$ grows when the graph is polynomially close to complete. For every fixed $β\in(0,1)$ and $G\sim G\!\left(n,1-n^{-(1-β)}\right)$, with high probability $\frac{2}β\le χ_m(G)\le \left\lfloor\frac{2}β\right\rfloor+5$. Thus $χ_m(G)=2/β+O(1)$. The lower bound is spectral, while the upper bound follows from multinomial anti-concentration and the Lov'asz Local Lemma.
