Associahedron chromatic number bounded simply and logarithmically
The chromatic number of the associahedron: simple and logarithmic
Data Structures and Algorithms
Summary
Coloring the faces of geometric shapes called associahedra is a challenging problem. The paper presents a simpler way to prove that the number of colors needed grows slowly with the size, specifically only about six times the logarithm of the size plus a constant. This is much more efficient than previous complicated proofs requiring many more colors. The authors hope that this simpler approach might help in finding even better colorings in future research.
What this means in practice
- •For graph algorithm designers: Use the tighter chromatic number bounds for associahedra to design more efficient graph coloring heuristics in complex geometric settings.
- •For optimization software engineers: Incorporate the simplified logarithmic coloring bounds to improve heuristics for constraint solving problems modeled with associahedra.
A theory result. No direct application yet.
Authors
Benjamin Aram Berendsohn, Jean Cardinal, John Iacono, László Kozma
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.