Cyclically Colored Triangulations: Enumeration and Connectedness of Reconfiguration Graphs

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.