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

  • Authors:

  • Ken-ichi Kawarabayashi;Kenta Ozeki

  • Affiliations:

  • National Institute of Informatics, 2-1-2, Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan;Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-0061, Japan

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2010

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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^.^2^0^7^2), 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.