OpenAI research collection · 1 manuscript · Source checked 2026-10-19
It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed 0< δ< 1/3, a deterministic polynomial-time reduction from 3SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size δ n otherwise, where n is the number of vertices.
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We prove that, for every fixed 0< δ< 1/3, 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.
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