Flips in colorful triangulations

TL;DR

This work investigates Hamiltonicity in flip graphs of triangulations under Ramsey-type colorability constraints on polygon vertices. It develops a general framework connecting colorful triangulations to tree-rotation Gray codes, proving that the colorful subgraph $\mathcal{F}_N$ has a Hamilton cycle for all $N\ge 8$ and that long coloring patterns $\alpha$ (with at least 10 color changes) likewise admit Hamilton cycles; it also provides an $O(1)$-average-time algorithm to enumerate a Hamilton path, based on a unified, constructive approach to tree rotations. The authors extend the framework to three colors via twists, establishing connectivity when $N$ is a multiple of 3, and present a unified proof for Hamiltonicity across $k$-ary trees, yielding efficient generation algorithms with $\mathcal{O}(k)$ average time per tree. They conclude with open questions about cycle spectra, higher-color models, and connections to pattern-avoiding permutations, highlighting rich combinatorial structure and practical enumeration methods for these constrained flip graphs.

Abstract

The associahedron is the graph $\mathcal{G}_N$ that has as nodes all triangulations of a convex $N$-gon, and an edge between any two triangulations that differ in a flip operation. A flip removes an edge shared by two triangles and replaces it by the other diagonal of the resulting 4-gon. In this paper, we consider a large collection of induced subgraphs of $\mathcal{G}_N$ obtained by Ramsey-type colorability properties. Specifically, coloring the points of the $N$-gon red and blue alternatingly, we consider only colorful triangulations, namely triangulations in which every triangle has points in both colors, i.e., monochromatic triangles are forbidden. The resulting induced subgraph of $\mathcal{G}_N$ on colorful triangulations is denoted by $\mathcal{F}_N$. We prove that $\mathcal{F}_N$ has a Hamilton cycle for all $N\geq 8$, resolving a problem raised by Sagan, i.e., all colorful triangulations on $N$ points can be listed so that any two cyclically consecutive triangulations differ in a flip. In fact, we prove that for an arbitrary fixed coloring pattern of the $N$ points with at least 10 changes of color, the resulting subgraph of $\mathcal{G}_N$ on colorful triangulations (for that coloring pattern) admits a Hamilton cycle. We also provide an efficient algorithm for computing a Hamilton path in $\mathcal{F}_N$ that runs in time $\mathcal{O}(1)$ on average per generated node. This algorithm is based on a new and algorithmic construction of a tree rotation Gray code for listing all $n$-vertex $k$-ary trees that runs in time $\mathcal{O}(k)$ on average per generated tree.

Figures (27)

• Figure 1: The graph of the 3-dimensional associahedron. The top edge of each triangulation is the outer edge that determines the root of the corresponding binary tree (see little arrow).
• Figure 2: Correspondence between flips in triangulations (top) and rotations in binary trees (bottom).
• Figure 3: Induced subgraphs of the associahedron ${\mathcal{G}}_5$ (top left) and ${\mathcal{G}}_6$ (bottom left) obtained for the coloring sequence ${\textcolor{red}{r}}{\textcolor{blue}{b}}{\textcolor{red}{r}}{\textcolor{blue}{b}}\cdots$ by forbidding monochromatic triangles. The triangulations with monochromatic triangles are still shown, but they are not part of the graphs ${\mathcal{F}}_5$ and ${\mathcal{F}}_6$ (top right and bottom right, respectively) and hence crossed out.
• Figure 4: Flip graph of colorful triangulations for the coloring sequence $\alpha=(4,4)$ (rrrrbbbb), which is isomorphic to the flip graph of $(3,3)$-combinations under adjacent transpositions. The black arrow in the leftmost triangulation is the ray that separates red from blue points, and the combination is obtained by reading the triangle types that intersect this ray from top to bottom (red=1, blue=0). The nodes of degree 1 and a Hamilton path in the flip graph are highlighted.
• Figure 5: Illustration of Theorem \ref{['thm:grid']} for the coloring sequence $\alpha=(4,1,3,1)$ (${\textcolor{red}{r}}{\textcolor{red}{r}}{\textcolor{red}{r}}{\textcolor{red}{r}}{\textcolor{blue}{b}}{\textcolor{red}{r}}{\textcolor{red}{r}}{\textcolor{red}{r}}{\textcolor{blue}{b}}$). The nodes of degree 1 in the flip graph are highlighted.

Theorems & Definitions (26)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7: MR1723053 for $k=2$; MR2474724 for $k\geq 3$
• Theorem 8
• Theorem 9
• Theorem 10: MR2426410