Uniform and balanced sampling of connected planar graph partitions is hard unless RP=NP, the flip walk mixes exponentially slowly on explicit triangulation families, and tractable cases include series-parallel and bounded-treewidth graphs.
On planar regular graphs degree three without Hamiltonian cycles
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abstract
Necessary condition to have Hamiltonian cycle in planar graph is given. Examples of regular planar graphs degree three without Hamiltonian cycle are built.
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Complexity and Geometry of Sampling Connected Graph Partitions
Uniform and balanced sampling of connected planar graph partitions is hard unless RP=NP, the flip walk mixes exponentially slowly on explicit triangulation families, and tractable cases include series-parallel and bounded-treewidth graphs.