$d$-to-$1$ Hardness of Coloring $4$-colorable Graphs with $O(1)$ colors

The d-to-1 conjecture of Khot asserts that it is NP-hard to satisfy an fraction of constraints of a satisfiable d-to-1 Label Cover instance, for arbitrarily small > 0. We prove that the d-to-1 conjecture for any fixed d implies the hardness of coloring a 3-colorable graph with C colors for arbitrarily large integers C. Earlier, the hardness of O(1)-coloring a 4-colorable graphs is known under the 2-to-1 conjecture, which is the strongest in the family of d-to-1 conjectures, and the hardness for 3-colorable graphs is known under a certain “fish-shaped” variant of the 2-to-1 conjecture. 2012 ACM Subject Classification Theory of computation → Computational complexity and cryptography

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