Sampling Eulerian orientations of triangular lattice graphs

Sampling Eulerian orientations of triangular lattice graphs

We consider the problem of sampling from the uniform distribution on the set of Eulerian orientations of subgraphs of the triangular lattice. Although it is known that this can be achieved in polynomial time for any graph, the algorithm studied here is more natural in the context of planar Eulerian graphs. We analyse the mixing time of a Markov chain on the Eulerian orientations of a planar gra...

Reliable Orientations of Eulerian Graphs

We present a characterization of Eulerian graphs that have a k-arc-connected orientation so that the deletion of any vertex results in a (k− 1)-arc-connected directed graph. This provides an affirmative answer for a conjecture of Frank [2]. The special case, when k = 2, describes Eulerian graphs admitting 2-vertexconnected orientations. This case was proved earlier by Berg and Jordán [1]. These...

Two-connected orientations of Eulerian graphs

A graph G = (V,E) is said to be weakly four-connected if G is 4-edgeconnected and G− x is 2-edge-connected for every x ∈ V . We prove that every weakly four-connected Eulerian graph has a 2-connected Eulerian orientation. This verifies a special case of a conjecture of A. Frank.

Asymptotic behavior of the number of Eulerian orientations of graphs

Sampling colourings of the triangular lattice

We show that the Glauber dynamics on proper 9-colourings of the triangular lattice is rapidly mixing, which allows for efficient sampling. Consequently, there is a fully polynomial randomised approximation scheme (FPRAS) for counting proper 9-colourings of the triangular lattice. Proper colourings correspond to configurations in the zero-temperature anti-ferromagnetic Potts model. We show that ...