关于可适应2-着色的临界窗口
On the Critical Window for Adaptable 2-Colorability
- University of Toronto（多伦多大学）
- School of Computer Science and Electrical Engineering, University of Ottawa（渥太华大学计算机科学与电气工程学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文确定了随机图可适应2-着色的尖锐阈值，刻画了决定性子的子图族，证明阈值之上存在交替长路径，给出与巨分量及2-SAT匹配的临界窗口界，并证明临界窗口下解空间连通。
AI中文摘要：
我们确定了随机图的可适应2-着色的尖锐阈值，该随机图配备了一个均匀随机的、不一定是正常的红/蓝边着色。为了实现这一点，我们刻画了一族子图及其边着色，这些子图及其边着色的包含或排除决定了可适应2-着色性。我们进一步证明，在阈值之上，会形成一条具有交替边颜色的长路径。我们利用这条路径来证明在超临界状态下存在这样的子图。然后，我们提供并证明了2-可适应着色临界窗口的对称界。特别地，我们证明的界与Erdős-Rényi随机图模型中巨分量的临界窗口以及随机2-SAT实例的可满足性的临界窗口相匹配。最后，我们证明在临界窗口之下，可适应2-着色的解空间保持连通，即可以通过一系列在O(log n)个顶点上不同的2-着色从一个可适应2-着色到达另一个可适应2-着色。
英文摘要：
We determine a sharp threshold for the adaptable 2-colorability of a random graph equipped with a uniformly random, not necessarily proper, red/blue coloring of the edges. To accomplish this, we characterize a family of subgraphs along with edge colorings whose inclusion or exclusion determines adaptable $2$-colorability. We further show that above the threshold, a long path with alternating edge colors is formed. We use this path to prove the existence of such a subgraph in the supercritical regime. We then provide and prove symmetric bounds on the critical window for $2$-adaptable colorability. Particularly, we prove bounds matching that of the critical windows for the giant component in the Erd$ő$s-R$é$nyi random graph model as well as the satisfiability of a random $2$-SAT instance. Finally, we show that below the critical window, the solution space of adaptable $2$-colorings remains connected, that is one can travel from one adaptable $2$-coloring to another by a sequence of $2$-colorings which differ on $O(\log{n})$ many vertices.
