关于高效图覆盖与引导随机游走
On efficient graph covers and steered random walks
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中文总结 AI 辅助
研究关于\(n\)顶点图顶点划分问题，核心方法是证明可将顶点划分为半径\(r = O(\log n)\)的部分，主要贡献是回答相关问题并改进\(\epsilon\)-引导随机游走最优覆盖时间上界，且证明该界对强顶点扩展图在常数因子内最优。
中文摘要 AI 辅助
我们证明，任何具有\(n\)个顶点的图的顶点可被划分为半径\(r = O(\log n)\)的部分，使得其闭邻域大小之和至多为\(4n\)。这回答了Bukh和Dubroff最近的一个问题，并直接改进了他们关于\(\epsilon\)-引导随机游走最优覆盖时间的上界。我们还证明，对于具有强顶点扩展的图，\(r\)的这个界在常数因子范围内是最优的。
英文摘要
We prove that the vertices of any $n$-vertex graph can be partitioned into pieces of radius $r = O(\log n)$ such that the sum of the sizes of their closed neighborhoods is at most $4n$. This answers a recent question of Bukh and Dubroff and directly yields an improvement to their upper bound on the optimal cover time of the $ε$-steered random walk. We also demonstrate that our bound on $r$ is best possible up to a constant factor for graphs with strong vertex expansion.
