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 1-1/k 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-O(1/k} of edges. Previously, only a hardness factor of 1-O(1/k^2) 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 1-1/k +2 ln k/k^2 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 1-1/k + O(ln k/ k^2) of edges of a k-colorable graph.
NP-hardness of additive approximation for generalized Turán problems
University of Science and Technology of China（中国科学技术大学）；East China Normal University（华东师范大学）；Shaanxi Normal University（陕西师范大学）；Tsinghua University（清华大学）；Fuzhou University（福州大学）
Maximum Partial List H-Coloring on P_5-free graphs in polynomial time
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Warsaw University of Technology（华沙理工大学）；Institute of Informatics, University of Warsaw（华沙大学信息学研究所）；Institute of Mathematical Sciences, Chennai（金奈数学科学研究所）；University of Bergen（卑尔根大学）；Discrete Mathematics Group, Institute for Basic Science (IBS)（基础科学研究院离散数学组）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）
