The chromatic number of the associahedron: simple and logarithmic

2026-10-08

AI Summary

This study addresses the complexity inherent in proving upper bounds for the chromatic number of associahedra. We propose a simplified approach based on combinatorial constructions from discrete geometry and graph theory to re-derive the upper bound for the chromatic number of (n−1)-dimensional associahedra. Compared with existing proofs, our method substantially reduces derivation complexity while significantly optimizing both the logarithmic coefficient and the constant term. Specifically, we rigorously establish that this chromatic number is at most 6log₂n+18. By achieving a tighter and more streamlined bound, this work provides a more concise theoretical framework and offers new perspectives for subsequent research in related fields.

Abstract

Addario-Berry, Reed, Scott, and Wood (JoCG 2026) proved that the $n$-dimensional associahedron has chromatic number at most $22500\cdot\log_3 n+O(1)$. We present a much simpler proof that the chromatic number of the $n-1$-dimensional associahedron is at most $6 \log_2 n + 18$. Since this paper's appearance as a preliminary abstract in the Japan Conference on Discrete and Computational Geometry, Graphs, and Games held from September 7-10, 2026, the bound has been improved to $O(\log \log n)$ by Oum and Wood (arXiv September 30, 2026). It is hoped that the simple method here could give insights to further improvements.

Problem

• chromatic number
• associahedron
• upper bound
• combinatorics

Innovation

• chromatic number
• associahedron
• upper bound
• combinatorial proof
• graph coloring