AI 中文总结
该研究针对循环着色凸多边形有效三角剖分的重构图，通过根边分解方法分别解决了3色情形的连通性猜想与计数递推，证明了4色及以上翻转图的连通性，建立了统一的结构分析框架。
AI 中文摘要
我们研究顶点用j≥3种颜色循环着色的凸多边形有效三角剖分的重构图的连通性与计数，其中每个三角形的三个顶点颜色两两不同。\n对于j=3的情况，我们证实了Acharya、Mütze和Verciani的一个猜想：证明了对所有k≥4，扭转图ℋ_{3k+2}是连通的，而ℋ₈和ℋ₁₁是不连通的。通过一种在状态空间中诱导笛卡尔积的着色根边分解方法，我们得到了T(3k)和T(3k+2)的耦合递推关系。对应的生成函数可化简为方程U(x)=1+xU(x)⁴，且两个连续族之间的差值由Raney数给出：T(3k+3)-T(3k+2)=R_{4,5}(k-1)。\n对于j≥4的情况，重构通过保持有效性的对角线翻转实现。我们将根边分解推广到所有满足N≢1(mod j)的容许类，对每个固定的j得到一个有限的函数方程代数系统。我们进一步证明，只要存在有效三角剖分，翻转图𝒢_N^{(j)}就是连通的。因此，根边分解为循环着色三角剖分的计数与重构提供了统一的结构框架。
英文摘要
We study the connectedness and enumeration of reconfiguration graphs of valid triangulations of convex polygons whose vertices are cyclically colored with $j \ge 3$ colors, where every triangle has vertices of three pairwise distinct colors. For $j = 3$, we settle a conjectural expectation of Acharya, Mütze, and Verciani: we prove that the twist graph $\mathcal{H}_{3k+2}$ is connected for every $k \ge 4$, whereas $\mathcal{H}_8$ and $\mathcal{H}_{11}$ are disconnected. Using a colored root-edge decomposition that induces Cartesian products in the state space, we obtain coupled recurrences for $T(3k)$ and $T(3k+2)$. The corresponding generating functions reduce to the equation $U(x) = 1 + xU(x)^4$, and the difference between the two consecutive families is given by the Raney number $T(3k+3) - T(3k+2) = R_{4,5}(k-1)$. For $j \ge 4$, reconfiguration is performed by validity-preserving diagonal flips. We extend the root-edge decomposition to all admissible classes $N \not\equiv 1 \pmod{j}$, obtaining, for each fixed $j$, a finite algebraic system of functional equations. We further prove that the flip graph $\mathcal{G}_N^{(j)}$ is connected whenever valid triangulations exist. Thus, the root-edge decomposition provides a unified structural framework for the enumeration and reconfiguration of cyclically colored triangulations.
Comments22 pages, 6 figures
