The Penrose-Kauffman polynomial equals a sum of chromatic polynomials for cubic graphs with perfect matchings on surfaces, and the Four Color Theorem is equivalent to 3-colorability of reduced alternating link diagrams without bigons.
Diestel, Graph Theory, 5th ed.
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The Penrose-Kauffman Polynomial
The Penrose-Kauffman polynomial equals a sum of chromatic polynomials for cubic graphs with perfect matchings on surfaces, and the Four Color Theorem is equivalent to 3-colorability of reduced alternating link diagrams without bigons.