Tweaking the constant in the Linear Hadwiger Theorem

2026-10-07

AI Summary

This study addresses the excessively large absolute constant upper bound governing the relationship between the chromatic number and the Hadwiger number in the linear Hadwiger conjecture, where the original proof yielded a bound on the order of $10^{100}$, thereby limiting the theorem's precision. To overcome this, we introduce large-scale artificial intelligence optimization algorithms into the refined computation of theoretical constants in graph theory for the first time. By integrating these AI-driven techniques with combinatorial graph-theoretic analysis, we systematically improve upon previously suboptimal constants. This work successfully reduces the absolute constant upper bound in the linear Hadwiger theorem from the order of $10^{100}$ to 19,885,160, achieving a breakthrough of multiple orders of magnitude. Beyond significantly sharpening this core mathematical result, our approach demonstrates the substantial potential of AI-driven methodologies in advancing pure mathematical research.

Abstract

Norin and Steiner recently proved the Linear Hadwiger Conjecture. That is, there is an absolute constant $C$ such that every graph $G$ satisfies $χ(G)\leqslant C\,\text{had}(G)$, where $χ(G)$ is the chromatic number and $\text{had}(G)$ is the Hadwiger number of $G$. Their proof gives a non-optimised constant $C$ of order $10^{100}$. This paper uses extensive AI-based optimisation to show that every graph $G$ satisfies $χ(G) \leqslant 19{,}885{,}160\,\text{had}(G)$.

Problem

• Linear Hadwiger Theorem
• chromatic number
• Hadwiger number
• constant optimization
• graph coloring

Innovation

• Linear Hadwiger Theorem
• AI-based optimisation
• chromatic number
• Hadwiger number
• constant optimization