On the Hardness of 4-Coloring a 3-Colorable Graph

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 n 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 4coloring 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 n (1) colors. Our proof technique also shows that there is an "0 > 0 such that it is NP-hard to legally 4-color even a (1 "0) fraction of the edges of a 3-colorable graph.