Enumeration of labeled 4-regular planar graphs
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In this extended abstract, we present the first combinatorial scheme for counting labeled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function counting labeled 4-regular planar graphs can be computed effectively as the solution of a system of equations. From here we can extract the coefficients by means of algebraic calculus. As a by-product, we can also compute the algebraic generating function counting labeled 3-connected 4-regular planar maps.
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CitacióNoy, M., Requile, C., Rue, J. Enumeration of labeled 4-regular planar graphs. "Electronic notes in discrete mathematics", 1 Agost 2017, vol. 61, p. 933-939.