Coloring k-colorable graphs using smaller palettes
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
On semidefinite programming relaxations for graph coloring and vertex cover
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Coloring k-colorable graphs using relatively small palettes
Journal of Algorithms
Exact complexity of exact-four-colorability
Information Processing Letters
Conditional hardness for approximate coloring
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Constructive generation of very hard 3-colorability instances
Discrete Applied Mathematics
Image data hiding schemes based on graph coloring
UIC'11 Proceedings of the 8th international conference on Ubiquitous intelligence and computing
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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 by work of Khanna, Linial and Safra (1993), but our proof is novel, as it does not rely on the PCP theorem, while the earlier one does. This highlights a qualitative difference between the known hardness result for coloring 3-colorable graphs and the factor n^{\epsilon} 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 Khanna, Linial and Safra). 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^{\Omega(1)} colors. Our proof technique also shows that there is a \eps_0 0 such that it is NP-hard to legally 4-color even a (1-\eps_0) fraction of the edges of a 3-colorable graph.