Theory Related Fields, 125 ( 2003 ), pp. Mossel, On the mixing time of a simple random walk on the super critical percolation cluster, Probab. Wormald, The mixing time of the giant component of a random graph, Random Structures Algorithms, 45 ( 2014 ), pp. Szemerédi, The longest path in a random graph, Combinatorica, 1 ( 1981 ), pp. Lei, The mixing time of the Newman-Watts small world, in Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '12, 2012, pp. All these results are asymptotically tight.ġ. Moreover, we show that if $G$ has bounded degeneracy, then typically the mixing time of the lazy random walk on $G^*$ is $O(\log^2 n)$. Given an $n$-vertex connected graph $G$, form a random supergraph $G^*$ of $G$ by turning every pair of vertices of $G$ into an edge with probability $\frac)$ and contains a path of length $\Omega(n)$, where for the last two properties we additionally assume that $G$ has bounded maximum degree. In this work, we study smoothed analysis on trees or, equivalently, on connected graphs. The main paradigm of smoothed analysis on graphs suggests that for any large graph $G$ in a certain class of graphs, perturbing slightly the edge set of $G$ at random (usually adding few random edges to $G$) typically results in a graph having much “nicer” properties.
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