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Revision #1 to TR19-116 | 5th May 2020 02:49
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#### $d$-to-$1$ Hardness of Coloring $3$-colorable Graphs with $O(1)$ colors

**Abstract:**
The $d$-to-$1$ conjecture of Khot asserts that it is NP-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 $3$-colorable graph with $C$ colors for arbitrarily large integers $C$.

Earlier, the hardness of $O(1)$-coloring a $4$-colorable graphs was 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 was known under a certain ``fish-shaped'' variant of the $2$-to-$1$ conjecture.

**Changes to previous version:**
The hardness is extended to $3$-colorable graphs by applying a subsequently discovered repeated arc digraph construction to reduce the chromatic number to 3. This version is the ICALP 2020 camera ready paper.

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TR19-116 | 9th September 2019 23:28
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#### $d$-to-$1$ Hardness of Coloring $4$-colorable Graphs with $O(1)$ colors

**Abstract:**
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.