Enumeration of plane triangulations with all vertices of degree $3$ or $6$ and a new characterization of akempic triangulations

TL;DR

The work addresses the enumeration of plane triangulations with all vertices of degree $3$ or $6$ and introduces a new characterization of akempic triangulations. It develops an index-vector framework $(K(q), M(q), S^{+}(q))$, leverages $\theta$-billiard sequences and Farey structures to propagate these indices, and incorporates symmetry and non-simple reductions to achieve complete enumeration results. The major contributions include a simple, new akempic criterion based on $\gcd$ conditions on index components, a streamlined proof of Mohar's enumeration, and a closed-form enumeration $d(n)$ in terms of divisor functions and congruence counts. Together, these results deepen the combinatorial understanding of colorings in constrained plane triangulations and offer a concise route to Mohar's theorems.

Abstract

Plane triangulations with all vertices of degree $3$ or $6$ are enumerated. A plane triangulation is said to be akempic if it has a $4$-colouring such that no two adjacent triangles have the same three colours and this colouring is not Kempe equivalent to any other colouring. Mohar (1985 and 1987) characterized and enumerated akempic triangulations with all vertices of degree $3$ or $6$. We give a new characterization of the akempic triangulations and a new proof of the Mohar enumeration theorem.

Figures (6)

• Figure 1: A triangulation $P_0$. Edges $g_{0}$, $g_{1}$, $g_{2}$ are of $0$, $1$, $2$-class, respectively. Each black edge is a left branch of the directed path $[A, 2]$. $S^{+}(0) = 3$. $S^{-}(0) = 4$.
• Figure 2: A triangulation $P_1$. Edges $g_{0}$, $g_{1}$, $g_{2}$ are of $0$, $1$, $2$-class, respectively. The black path $A D$ is of $2$-class. The black path $DH$ is of $0$-class. $S^{+}(1) = [A, 1](g_2) = 0$. $S^{-}(1) = [B, 1](h) = 1$.
• Figure 3: A triangulation $P_2$. Edges $g_{0}$, $g_{1}$, $g_{2}$ are of $0$, $1$, $2$-class, respectively. Black paths are of $0$-class. $S^{+}(2) = [A, 2](e) = 1$. $S^{-}(2) = 3$.
• Figure 4: A triangulation $P_{q}$ with the index $(K(q), M(q), S^{+}(q)) = (3, 3, 0)$ ($(2, 3, 2)$, respectively). $S^{+}(q) + S^{-}(q) = M(q)$. The line containing the inner path of $q$-class is the axis of symmetry of $P_{q} - v_{1}v_{2}$, where $v_{0}v_{1}v_{2}v_{3}$ is the exterior path of $q$-class.
• Figure 5: A triangulation $P_{q}$ with the index $(K(q), M(q), S^{+}(q)) = (3, 4, 1)$. $S^{+}(q) + S^{-}(q) \neq M(q)$. The line containing the path of $q$-class is not the axis of symmetry of $P_{q} - v_{1}v_{2}v_{3}$, where $v_{0}v_{1}v_{2}v_{3}v_{4}$ is the exterior path of $q$-class.

Theorems & Definitions (44)

• Proposition 1.1
• Proposition 1.2
• Proposition 1.3
• Proposition 1.4
• Proposition 1.5
• Proposition 1.6
• Theorem 1.1
• Theorem 1.2
• Proposition 1.7
• Remark 1.1