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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, while the one in  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 .
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 ). 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
Guruswami, Venkatesan and Khanna, Sanjeev, "On the hardness of 4-coloring a 3-colorable graph" (2000). Departmental Papers (CIS). Paper 75.
Date Posted: 11 March 2005