On the hardness of 4-coloring a 3-collorable 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, but our proof is novel as it does not rely on the PCP theorem. This highlights a qualitative difference between the known hardness result for coloring 3-colorable graphs and the factor n^ε hardness for approximating the chromatic number of general graphs, as the latter result is known to imply (some form of) PCP theorem. 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. 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^Ω(1) colours. 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