Cm:
TWO LINEAR TIMES ALGORITHMS FOR THE FIVE-COLOR OF A PLANAR GRAPH David Matula, Yossi Shiloah, Robert Taryan
(Just google it and you will find the pdf document).
So this is an article about 5-coloring a planar graph in O (n) time, but it starts with a simple description of the 6-coloring algorithm. Here's an important extract (sorry for formatting, this is just cleaning the PDF):
6 . n- G , 6- G. 1. [ .] j, 0-j - n - 1, < < G j. - 2. [ .] = n, n - 1, n * - 1,.,, 1 j j t/i. vi j-. U, tli G , f, u jr u j9 - 1 . 3. [ .] = 1,2,.,, n, t) ( ), , t) i, .