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.

翻译：关联多面体图 $\mathcal{G}_N$ 的节点是凸 $N$ 边形的所有三角剖分，若两个三角剖分可通过一次翻转操作相互转换，则其间存在一条边。翻转操作指移除两个三角形共享的一条边，并将其替换为所形成的四边形的另一条对角线。本文考虑通过拉姆齐型可着色性质得到的一大类 $\mathcal{G}_N$ 的诱导子图。具体而言，将 $N$ 边形的顶点交替着红色与蓝色，我们仅考虑彩色三角剖分，即每个三角形都包含两种颜色顶点的三角剖分，亦即禁止单色三角形。由此得到的 $\mathcal{G}_N$ 在彩色三角剖分上的诱导子图记为 $\mathcal{F}_N$。我们证明对于所有 $N\geq 8$，$\mathcal{F}_N$ 均存在哈密顿环，这解决了 Sagan 提出的一个问题，即所有 $N$ 个顶点的彩色三角剖分可以排列成一个列表，使得列表中任意两个循环相邻的三角剖分仅通过一次翻转即可相互转换。事实上，我们证明对于 $N$ 个顶点的任意固定着色模式（要求颜色变化次数至少为 10），对应的 $\mathcal{G}_N$ 在彩色三角剖分（针对该着色模式）上的诱导子图均存在哈密顿环。我们还提出了一种计算 $\mathcal{F}_N$ 中哈密顿路径的高效算法，该算法平均每生成一个节点的时间复杂度为 $\mathcal{O}(1)$。此算法基于一种新的、可算法化实现的树旋转格雷码构造，该格雷码可用于列出所有 $n$ 个顶点的 $k$ 叉树，且平均每生成一棵树的时间复杂度为 $\mathcal{O}(k)$。