Stefan Felsner ; Daniel Heldt - Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs

dmtcs:1376 - Discrete Mathematics & Theoretical Computer Science, February 3, 2017, Vol. 18 no. 3 - https://doi.org/10.46298/dmtcs.1376
Mixing Times of Markov Chains on Degree Constrained Orientations of Planar GraphsArticle

Authors: Stefan Felsner ; Daniel Heldt

    We study Markov chains for $\alpha$-orientations of plane graphs, these are orientations where the outdegree of each vertex is prescribed by the value of a given function $\alpha$. The set of $\alpha$-orientations of a plane graph has a natural distributive lattice structure. The moves of the up-down Markov chain on this distributive lattice corresponds to reversals of directed facial cycles in the $\alpha$-orientation. We have a positive and several negative results regarding the mixing time of such Markov chains. A 2-orientation of a plane quadrangulation is an orientation where every inner vertex has outdegree 2. We show that there is a class of plane quadrangulations such that the up-down Markov chain on the 2-orientations of these quadrangulations is slowly mixing. On the other hand the chain is rapidly mixing on 2-orientations of quadrangulations with maximum degree at most 4. Regarding examples for slow mixing we also revisit the case of 3-orientations of triangulations which has been studied before by Miracle et al.. Our examples for slow mixing are simpler and have a smaller maximum degree, Finally we present the first example of a function $\alpha$ and a class of plane triangulations of constant maximum degree such that the up-down Markov chain on the $\alpha$-orientations of these graphs is slowly mixing.


    Volume: Vol. 18 no. 3
    Section: Graph Theory
    Published on: February 3, 2017
    Accepted on: December 5, 2016
    Submitted on: February 3, 2017
    Keywords: Mathematics - Combinatorics,Mathematics - Probability,05C10, 05C15, 60J05, 68Q87, 68R10

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