A Lovász-Kneser theorem for triangulations

TL;DR

This work identifies the chromatic number of the Kneser graph on triangulations of a convex $n$-gon, proving $\chi(\mathrm{KG}(\mathcal{T}_n)) = n-2$ by tying triangulations to the associahedron via the GKZ secondary polytope and applying a topological bound for Kneser graphs of polytopes. The authors develop a general polytope-based framework (bilinear form, facet vectors, and hemisphere cover) that yields $\chi(\mathrm{KG}(P)) \ge d+1$ for a $d$-dimensional polytope, then realize $\mathrm{Assoc}_{n-3}$ as $\Sigma_Q$ to obtain the lower bound. They also establish stability results, showing there exists a large induced subgraph $\mathrm{KG}(\mathcal{T}_n^{(3)})$ with $|\mathcal{T}_n^{(3)}|=F_{2n-5}$ and the same chromatic number, using a swapping relation and a 3-parenthesization construction. In higher dimensions, the paper discusses limits of the approach (via totally splittable polytopes and the permutohedron), and poses open problems on independence numbers and Kneser hypergraphs, highlighting ongoing connections between triangulation combinatorics, polytope geometry, and topological methods.

Abstract

We show that the Kneser graph of triangulations of a convex $n$-gon has chromatic number $n-2$.

Figures (7)

• Figure 1: $\floor{n/2}$ pairwise disjoint zig-zag triangulations of an $n$-gon for $4 \leq n \leq 8$.
• Figure 2: The associahedron $\mathrm{Assoc}_3$ and a proper $4$-coloring of $\mathop{\mathrm{KG}}\nolimits(\mathcal{T}_6)$.
• Figure 3: A $2$-coloring of $\mathrm{Diag}_6$ demonstrating that $\mathop{\mathrm{cd}}\nolimits_2(\mathcal{T}_6) = 0$.
• Figure 4: \ref{['fig:p1']} An element of $X_Q$, used in the proof of \ref{['lem:vec']}. \ref{['fig:p2']} Points used in the proof of \ref{['lem:circ']}.
• Figure 5: \ref{['fig:zswap']} An illustration of the argument that $\chi(\mathop{\mathrm{KG}}\nolimits(\mathcal{T}_6 \setminus \set{Z})) = 4$. \ref{['fig:gswap']} An illustration of the $\rightsquigarrow$ relation. Here $\set{i,j} \rightsquigarrow \set{i',j'}$.

Theorems & Definitions (14)

• proof : Proof of \ref{['lem:polybu']}
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• proof : Proof of \ref{['thm:main']}
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• proof : Proof of \ref{['thm:star']}