Coloring 3-colorable graphs with o(n 1/5 ) colors

Abstract

Recognizing 3-colorable graphs is one of the most famous NP-complete problems (Garey, John- son, and Stockmeyer STOC'74). The problem of coloring 3-colorable graphs in polynomial time with as few colors as possible has been intensively studied: O(n 1/2 ) colors (Wigderson STOC'82), someone not from computer science, use the example of 2-coloring versus 3-coloring: suppose there is too much fighting in a class, and you want to split it so that no enemies end up in the same group. First you try with a red and a blue group. Put someone in the red group, and everyone he dislikes in the blue group, everyone they dislike in the red group, and so forth. This is an easy systematic approach. Digging a bit deeper, if something goes wrong, you have an odd cycle, and it is easy to see that if you have a necklace with an odd number of red and blue beads, then the colors cannot alternate perfectly. This illustrates both ecient algorithms and the concept of a witness. Knowing that red and blue do not suce, we might