On the Enumeration of Four-Colored Maps
W. T. Tutte · SIAM Journal on Applied Mathematics · 1969
A nonnegative integer called the “rank” will be associated with each four-coloring of a rooted trivalent planar map. Our two main results are asymptotic expressions for the average number of four-colorings of ranks 0 and 1 for maps of $2n$ vertices. For rank 0 the average is asymptotically \[ 8(3\pi n)^{ - 1/2} (32/27)^n, \]and for rank 1, \[ 8(3\pi )^{ - 1/2} (4 /\pi - 1) n^{1/2} (32/27)^n . \]