Finding a proper coloring of a t-colorable G with t colors is a classic NP-hard problem when t >= 3. In this talk, we investigate the approximate coloring problem in which the objective is to find a proper c-coloring of G where c >= t. We show that for all t >= 3, it is NP-hard to find a c-coloring when c <= 2t-2. In the regime where t is small, this improves on the previously best known hardness result of c <= max{2t- 5, t + 2*floor(t/3) - 1} (Garey and Johnson 1976; Khanna, Linial, Safra, 1993; Guruswami, Khanna, 2000). For example, we show that 6-coloring a 4-colorable graph is NP-hard, improving on the NP-hardness of 5-coloring a 4-colorable graph.

We also generalize this to related problems on the strong coloring of hypergraphs. A k-uniform hypergraph H is t-strong colorable (where t >= k) if there is a t-coloring of the vertices such that no two vertices in each hyperedge of H have the same color. We show that if t = ceiling(3k/2), then it is NP-hard to find a 2-coloring of the vertices of H such that no hyperedge is monochromatic.

We establish the NP-hardness of these problems via reduction from the hardness of the Label Cover problem. In this reduction, we utilize a `dictatorship test' gadget graph similar to a long code test. By combinatorially classifying all possible colorings of this graph, we can infer labels to provide to the label cover problem. This approach generalizes the ``weak polymorphism" framework of (Austrin, Guruswami, Hastad, 2014), though interestingly our results are "PCP-free" in that they do not require any approximation gap in the starting Label Cover instance.

*This work is joint with Venkatesan Guruswami.*