# On the hardness of 4-coloring a 3-collorable graph

@article{Guruswami2000OnTH, title={On the hardness of 4-coloring a 3-collorable graph}, author={Venkatesan Guruswami and Sanjeev Khanna}, journal={Proceedings 15th Annual IEEE Conference on Computational Complexity}, year={2000}, pages={188-197} }

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/sup /spl epsiv// 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…

## 26 Citations

The hardness of 3-uniform hypergraph coloring

- Computer Science, MathematicsThe 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
- 2002

It is proved that coloring a 3-uniform 2-colorable hypergraph with any constant number of colors is NP-hard, and a certain maximization variant of the Kneser conjecture is proved, namely that any coloring of theKneser graph by fewer colors than its chromatic number, has 'many' non-monochromatic edges.

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Two new coloring results are obtained by combining the coloring algorithm of Karger, Motwani, and Sudan, the combinatorial coloring algorithms of Blum and an extension of a technique of Alon and Kahale for finding relatively large independent sets in graphs that are guaranteed to have very largeindependent sets.

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- Mathematics, Computer ScienceProceedings 41st Annual Symposium on Foundations of Computer Science
- 2000

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- Computer Science, MathematicsJ. Univers. Comput. Sci.
- 2006

It is shown that it is DP-complete to decide whether or not a given graph can be colored with exactly four colors, where DP is the second level of the boolean hierarchy.

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- Computer Science, MathematicsDiscret. Appl. Math.
- 2008

This work proposes a rather systematic or constructive algorithm that repeats the embedding of 4-critical graphs to arbitrarily generate large extraordinarily hard 3-colorability instances and demonstrates experimentally that the computational cost to solve these instances is of an exponential order of the number of vertices.

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The AprxColoring problem is studied and tight bounds on generalized noise-stability quantities are extended, which extend the recent work of Mossel, O'Donnell, and Oleszkiewicz and should have wider applicability.

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