The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
The Size of the Giant Component of a Random Graph with a Given Degree Sequence
Combinatorics, Probability and Computing
The Cover Time of Random Regular Graphs
SIAM Journal on Discrete Mathematics
The cover time of the giant component of a random graph
Random Structures & Algorithms
The scaling window for a random graph with a given degree sequence
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
A critical point for random graphs with a given degree sequence
Random Structures & Algorithms
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We consider random walks on two classes of random graphs and explore the likely structure of the the set of unvisited vertices (or vacant set). Let Γ(t) be the subgraph induced by the vacant set. We show that for random graphs Gn,p above the connectivity threshold, and for random regular graphs Gr, for constant r ≥ 3, there is a phase transition in the sense of the well-known Erdős-Renyi phase transition. Thus for t ≤ (1 − ε)t* we have a unique giant plus components of size O(log n) and for t ≥ (1 + ε)t* we have only components of size O(log n). In the case of Gr we describe the likely degree sequence and structure of the small (O(log n)) size components.