For non-bipartite planar cubic graphs with 2n vertices, the Heawood system has rank n+1, giving a defining set of n-1 vertices and at most 3 times 2^{n-1} Tait colorings; circular ladders have 2^n+8 colorings for even n and 2^n-2 for odd n.
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The Heawood approach to Tait colorings and defining vertex sets
For non-bipartite planar cubic graphs with 2n vertices, the Heawood system has rank n+1, giving a defining set of n-1 vertices and at most 3 times 2^{n-1} Tait colorings; circular ladders have 2^n+8 colorings for even n and 2^n-2 for odd n.