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TR19-116 | 9th September 2019 23:28

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

TR19-116
Authors: Venkatesan Guruswami, Sai Sandeep
Publication: 9th September 2019 23:29
The $d$-to-$1$ conjecture of Khot asserts that it is hard to satisfy an $\epsilon$ fraction of constraints of a satisfiable $d$-to-$1$ Label Cover instance, for arbitrarily small $\epsilon > 0$. We prove that the $d$-to-$1$ conjecture for any fixed $d$ implies the hardness of coloring a $4$-colorable graph with $C$ colors for arbitrarily large integers $C$. Earlier, this implication was only known under the $2$-to-$1$ conjecture, which is the strongest in the family of $d$-to-$1$ conjectures.