Even triangulations of 𝑆³ and the coloring of graphs
Jacob Eli Goodman, Hironori Onishi · Transactions of the American Mathematical Society · 1978
A simple necessary and sufficient condition is given for the vertices of a graph, planar or not, to be properly four-colorable. This criterion involves the notion of an “even” triangulation of S 3 {S^3} and generalizes, in a natural way, a corresponding criterion for the three-colorability of planar graphs.