Randomly Coloring Regular Bipartite Graphs and Graphs with Bounded Common Neighbors
碩士 === 國立臺灣大學 === 資訊工程學研究所 === 97 === Let G be an n-node graph with maximum degree △. The Glauber dynamics for G, defined by Jerrum, is a Markov chain over the k-colorings of G. Many classes of G on which the Glauber dynamics mixes rapidly have been identified. Recent research efforts focus on the i...
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| Format: | Others |
| Language: | en_US |
| Published: |
2009
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| Online Access: | http://ndltd.ncl.edu.tw/handle/00439599311820630491 |
| Summary: | 碩士 === 國立臺灣大學 === 資訊工程學研究所 === 97 === Let G be an n-node graph with maximum degree △. The Glauber dynamics for G, defined by Jerrum, is a Markov chain over the k-colorings of G. Many classes of G on which the Glauber dynamics mixes rapidly have been identified. Recent research efforts focus on the important case that △≧d log_2 n holds for some sufficiently large constant d. We add the following new results along this direction, where ε can be any constant with 0 < ε < 1.
1. Let α≒1.645 be the root of (1-e^{{-1}/x})^2+ 2x
e^{-1/x}=2. If G is regular and bipartite and k≥(α+ε) △， then the mixing time of the Glauber dynamics for G is O(nlog n).
2.Let β≒1.763 be the root of x=e^{1/x}. If the number
of common neighbors for any two adjacent nodes of G is at most ε^{1.5}Delta/360e且k≥(1+ε)β△， then the mixing time of the Glauber dynamics is O(nlog n).
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