Graph Colouring Is Hard on Average for Polynomial Calculus and Nullstellensatz

Jonas Conneryd; Susanna F. de Rezende; Jakob Nordström; Shuo Pang; Kilian Risse

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arXiv · 2503.17022

Graph Colouring Is Hard on Average for Polynomial Calculus and Nullstellensatz

Jonas Conneryd·Susanna F. de Rezende·Jakob Nordström·Shuo Pang·Kilian Risse

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Abstract

We prove that polynomial calculus (and hence also Nullstellensatz) over any field requires linear degree to refute that sparse random regular graphs, as well as sparse Erd\H{o}s-R\'{e}nyi random graphs, are $3$-colourable. Using the known relation between size and degree for polynomial calculus proofs, this implies strongly exponential lower bounds on proof size.

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Jonas Conneryd, Susanna F. de Rezende, Jakob Nordström, Shuo Pang, Kilian Risse. 2025-03-21. Graph Colouring Is Hard on Average for Polynomial Calculus and Nullstellensatz. https://arxiv.org/abs/2503.17022

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