On the conditional hardness of coloring a 4-colorable graph with super-constant number of colors
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Dominating set based exact algorithms for 3-coloring
Information Processing Letters
Linear index coding via semidefinite programming
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
| Hi-index | 0.00 |
We give a new proof showing that it is NP-hard to color a 3-colorable graph using just 4 colors. This result is already known , [S. Khanna, N. Linial, and S. Safra, Combinatorica, 20 (2000), pp. 393--415], but our proof is novel because it does not rely on the PCP theorem, while the known 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 [M. Bellare, O. Goldreich, and M. Sudan, SIAM J. Comput., 27 (1998), pp. 805--915].Another aspect in which our proof is novel is in its use of the PCP theorem to 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 an $\varepsilon_0 0$ such that it is NP-hard to legally 4-color even a $(1-\varepsilon_0)$ fraction of the edges of a 3-colorable graph.