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

Recognizing 3-colorable graphs is one of the most famous NP-complete problems [Garey, Johnson, 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(n1/2) colors [Wigderson STOC’82], Õ(n2/5) colors [Blum STOC’89], Õ(n3/8) colors [Blum FOCS’90], O(n1/4) colors [Karger, Motwani, Sudan FOCS’94], Õ(n3/14) = O(n0.2142) colors [Blum and Karger IPL’97], O(n0.2111) colors [Arora, Chlamtac, and Charikar STOC’06], and O(n0.2072) colors [Chlamtac FOCS’07]. Recently the authors got down to O(n0.2049) colors [FOCS’12]. In this paper we get down to O(n0.19996) = o(n1/5) colors. Since 1994, the best bounds have all been obtained balancing between combinatorial and semi-definite approaches. We present a new combinatorial recursion that only makes sense in collaboration with semi-definite programming. We specifically target the worst-case for semidefinite programming: high degrees. By focusing on the interplay, we obtained the biggest improvement in the exponent since 1997. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems – Computations on discrete structures, G.2.2 Graph Theory – Graph algorithms