Linear-time algorithms to color topological graphs

We describe a linear-time algorithm for 4-coloring planar graphs. We indeed give an O(V + E + |χ| + 1)-time algorithm to C-color V -vertex E-edge graphs embeddable on a 2-manifold M of Euler characteristic χ where C(M) is given by Heawood’s (minimax optimal) formula. Also we show how, in O(V + E) time, to find the exact chromatic number of a maximal planar graph (one with E = 3V − 6) and a coloring achieving it. Finally, there is a linear-time algorithm to 5-color a graph embedded on any fixed surface M except that an M-dependent constant number of vertices are left uncolored. All the algorithms are simple and practical and run on a deterministic pointer machine, except for planar graph 4-coloring which involves enormous constant factors and requires an integer RAM with a random number generator. All of the algorithms mentioned so far are in the ultra-parallelizable deterministic computational complexity class “NC.” We also have more practical planar 4-coloring algorithms that can run on pointer machines in O(V log V ) randomized time and O(V ) space, and a very simple deterministic O(V )-time coloring algorithm for planar graphs which conjecturally