Departmental Papers (CIS)

Date of this Version

July 2000

Document Type

Conference Paper


Copyright 2000 IEEE. Reprinted from Proceedings of the 15th Annual IEEE Conference on Computational Complexity 2000, pages 188-197.
Publisher URL:

This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to By choosing to view this document, you agree to all provisions of the copyright laws protecting it.


We give a new proof showing that it is NP-hard to color a 3-colorable graph using just four colors. This result is already known [18], but our proof is novel as it does not rely on the PCP theorem, while the one in [18] does. This highlights a qualitative difference between the known hardness result for coloring 3-colorable graphs and the factor nε hardness for approximating the chromatic number of general graphs, as the latter result is known to imply (some form of) PCP theorem [3].

Another aspect in which our proof is different is that using the PCP theorem we can show that 4-coloring of 3-colorable graphs remains NP-hard even on bounded-degree graphs (this hardness result does not seem to follow from the earlier reduction of [18]). We point out that such graphs can always be colored using O(1) colors by a simple greedy algorithm, while the best known algorithm for coloring (general) 3-colorable graphs requires nΩ(1) colors. Our proof technique also shows that there is an ε0 > 0 such that it is NP-hard to legally 4-color even a (1 - ε0) fraction of the edges of a 3-colorable graph.


computational complexity, graph coloring, 3-colorable graph, 4-coloring, NP-hard, bounded-degree graphs, chromatic number, hardness



Date Posted: 11 March 2005