A simple algorithm for 4-coloring 3-colorable planar graphs

Graph coloring for 3-colorable graphs receives very much attention by many researchers in theoretical computer science. Deciding 3-colorability of a graph is a well-known NP-complete problem. So far, the best known polynomial approximation algorithm achieves a factor of O(n0.2072), and there is a strong evidence that there would be no polynomial time algorithm to color 3-colorable graphs using at most c colors for an absolute constant c.In this paper, we consider 3-colorable PLANAR graphs. The Four Color Theorem (4CT) (Appel and Haken (1977) [1], , Appel et al. (1977) [2], , Robertson et al. (1997) [14]) gives an O(n2) time algorithm to 4-color any planar graph. However the current known proof for the 4CT is computer assisted. In addition, the correctness of the proof is still lengthy and complicated.We give a very simple O(n2) algorithm to 4-color 3-colorable planar graphs. The correctness needs only a 2-page proof.