有色随机图中的大单色分量
Large Monochromatic Components in Colored Random Graphs
AI总结：
研究随机图边着色中最大单色连通分量大小，通过分析顶点划分上的连通结构，证明\(G(n,p)\)的\(2\)边着色中单色连通分量阶至少为\(n - \Theta(ne^{-h})\)，扩展到\(3\)边着色，二分图情况也有类似结论，且各结论在常数因子上最优。
AI中文摘要：
我们研究了随机图的任何边着色中必然出现的最大单色连通分量的大小。设\(G\sim G(n,p)\)，其中\(p\gg 1/n\)且\(p = o(1)\)，并记\(np = he^h\)。我们证明，以高概率，\(G\)的每一种\(2\)边着色都包含一个阶至少为\(n - \Theta(ne^{-h})\)的单色连通分量，且构造的着色表明此界在常数因子上是最优的。我们将此结果扩展到三种颜色，对于\(p\gg 1/n\)且\(p = o(1)\)，以高概率\(G\)的每一种\(3\)边着色都包含一个大小至少为\(\frac{n}{2}-\Theta(1/p)\)的单色连通分量，且此估计在常数因子上也是紧的。在二分图\(G\sim G(n,n,p)\)的情况下，在相同的\(p\)假设下，我们证明了类似的结论：以高概率，每一种\(2\)边着色都包含两个单色分量，其并集覆盖除\(\Theta(ne^{-h})\)个顶点外的所有顶点，且此界是渐近紧的。我们的方法是基本的，基于分析适当平衡的顶点划分上的大连通结构。
英文摘要：
We study the size of the largest monochromatic connected component that must appear in any edge-coloring of a random graph. Let $G\sim G(n,p)$ with $p\gg 1/n$ and $p=o(1)$, and write $np=he^h$. We show that, with high probability, every $2$-edge-coloring of $G$ contains a monochromatic connected component of order at least $n-Θ(ne^{-h})$. Moreover, we construct colorings showing that this bound is best possible up to constant factors. We extend this result to three colors: for $p\gg 1/n$ and $p=o(1)$, with high probability every $3$-edge-coloring of $G$ contains a monochromatic connected component of size at least $\frac{n}{2}-Θ(1/p)$, and this estimate is again tight up to constant factors. In the bipartite setting $G\sim G(n,n,p)$, under the same assumptions on $p$, we prove an analogous statement: with high probability, every $2$-edge-coloring contains two monochromatic components whose union covers all but $Θ(ne^{-h})$ vertices, and this bound is asymptotically sharp. Our approach is elementary and is based on analyzing large connected structures across suitably balanced vertex partitions.
