OpenAI · 106-01 · 2026-09-24 · Theoretical computer science
Hardness of finding large independent sets in three-colorable graphs
preprints/Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026
Family 106. Hardness of coloring three-colorable graphs.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
It is NP-hard to color a three-colorable graph using any fixed number of colors. More strongly, for every fixed , a deterministic polynomial-time reduction from SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size otherwise, where is the number of vertices.
We prove that, for every fixed , it is NP-hard to distinguish three-colorable graphs from graphs in which every independent set has fewer than δ times the number of vertices. Consequently, for every fixed integer c ≥ 3, finding a proper c-coloring of a three-colorable graph is NP-hard.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 106: Hardness of coloring three-colorable graphs. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.