We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a k-colorable graph with k colors so that a maximum fraction of edges are properly colored (i.e., their endpoints receive different colors). A random k-coloring properly colors an expected fraction 1−1k of edges. We prove that given a graph promised to be k-colorable, it is NP-hard to find a k-coloring that properly colors more than a fraction 1−1(33k) of edges. Previously, only a hardness factor of 1−O(1k2) was known. Our result pins down the correct asymptotic dependence of the approximation factor on k. Along the way, we prove that approximating the Maximum 3-colorable subgraph problem within a factor greater than 32/33 is NP-hard.

Using semidefinite programming, it is known that one can do better than a random coloring and properly color a fraction 1−1k+k22lnk of edges in polynomial time. We show that, assuming the 2-to-1 conjecture, it is hard to properly color(using k colors) more than a fraction 1−1k+Ok2lnk of edges of a k-colorable graph.