A logarithmic bound for the chromatic number of the associahedron

TL;DR

This paper proves that the chromatic number of the flip-graph of triangulations of an $n$-gon, the associahedron $\mathcal{A}_n$, grows only logarithmically with $n$, hence resolving a conjecture of Fabila-Monroy et al. It achieves this by constructing a multi-component colouring that tracks several carefully chosen invariants under edge flips, including edge scales, quadrilateral types, and parity-based refinements, and then combining these with a colouring of small associahedra. The main result provides an explicit bound of the form $\chi(\mathcal{A}_n)=O(\log n)$, improving the previous $O(n/\log n)$ bound and advancing understanding of chromatic properties in flip graphs of combinatorial polytopes. The techniques, involving scale-based analysis and dual-tree decompositions, may have broader applicability to related geometric combinatorics problems.

Abstract

We show that the chromatic number of the $n$-dimensional associahedron grows at most logarithmically with $n$, improving a bound from and proving a conjecture of Fabila-Monroy et al. (2009).

Figures (9)

• Figure 1: Portions of two adjacent triangulations of an $n$-gon.
• Figure 2: Writing $\sigma=\max(\sigma_{AB},\sigma_{BC},\sigma_{CD})$, the subfigures correspond to possible configurations arising in the proof of Proposition \ref{['prop:2_same']}.
• Figure 3: The dual trees of an $8$-gon and of a sub-triangulation of a $12$-gon
• Figure 4: In both subfigures, left-turn edges are red and right-turn edges are blue.
• Figure 5: The left-turn and right-turn labels near quadrilateral $ABCD$ in $T$ and $T'$: case (a). Here $(g(x),d(x))=(a,b)$, $(g(v),d(v))=(a+1,b)$ and $(g(u),d(u))=(a+1,b+1)$.

Theorems & Definitions (14)

• Theorem 1
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Proposition 6