论文标题
密集的随机块图的色数
The Chromatic Number of Dense Random Block Graphs
论文作者
论文摘要
图$ g $的色数$χ（g）$，即为$ g $的顶点着色所需的最小颜色，因此没有两个相邻的顶点分配给相同的颜色，是经典且经过广泛研究的参数。在这里，我们考虑$ g $是随机块图，也称为随机块模型的情况。顶点集将$ k \分为$ k \ in \ mathbb {n} $ parts $ v_1，\ dotsc，v_k $，对于每个$ 1 \ le i \ le i \ le i \ le j \ le j \ le j \ le k $，两个vertices $ u \ in v_i in v_i，v \ in v_j $ in v_j $中的v_j $ in Edge in Edge in Edge in Edecrability $ p_ $ p_} in Indepency $ p_} in Indemability $ p_}。我们的主要结果销售了典型的渐近值$χ（g）$，并在最佳着色中建立了颜色类别的分布。我们发现，与我们模型中$ k = 1 $相对应的二项式随机图$ g（n，p）$形成鲜明对比，在我们的模型中，（几乎）最佳最佳着色基本上与独立数字一致的颜色类别的平均大小相吻合，块模型揭示了一个更加多样化的图片：“总体上的平均尺寸”，这些尺寸是conve yess and conve conve and conve conve and conve and conve convers cons and conve and conve conver的尺寸。与每个$ v_i $，$ 1 \ le i \ le K $的交集。
The chromatic number $χ(G)$ of a graph $G$, that is, the smallest number of colors required to color the vertices of $G$ so that no two adjacent vertices are assigned the same color, is a classic and extensively studied parameter. Here we consider the case where $G$ is a random block graph, also known as the stochastic block model. The vertex set is partitioned into $k\in\mathbb{N}$ parts $V_1, \dotsc, V_k$, and for each $1 \le i\le j\le k$, two vertices $u \in V_i, v\in V_j$ are connected by an edge with some probability $p_{ij} \in (0,1)$ independently. Our main result pins down the typical asymptotic value of $χ(G)$ and establishes the distribution of the sizes of the color classes in optimal colorings. We discover that in contrast to the case of a binomial random graph $G(n,p)$, that corresponds to $k=1$ in our model, where the average size of a color class in an (almost) optimal coloring essentially coincides with the independence number, the block model reveals a more diverse picture: the "average" class in an optimal coloring is a convex combination of several types of independent sets that vary in total size as well as in the size of their intersection with each $V_i$, $1\le i \le k$.
