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

Abstract

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 ( n 0.2072 ) , 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 ( n 2 ) 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 ( n 2 ) algorithm to 4-color 3-colorable planar graphs. The correctness needs only a 2-page proof.