Flips in colorful triangulations

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.