Transitions of Hexangulations on the Sphere

Abstract

A hexangulation G is a 2-connected simple plane graph such that each face of G is bounded by a 6-cycle. It was recently proved that any two hexangulations with the same number of vertices can be transformed into each other by three specifically defined transformations ${\mathbb{A}, \mathbb{B}}$ and ${\mathbb{C}}$. We prove that any two hexangulations G and G′ with bipartitions {B, W} and {B′, W′}, respectively, such that |B| = |B′| and |W| = |W′|, can be transformed into each other by successive applications of operations in ${\{\mathbb{A}, \mathbb{B}^2, \mathbb{C}\}}$. Moreover, we completely describe the role of the four operations ${\mathbb{A}, \mathbb{B}, \mathbb{B}^2}$ and ${\mathbb{C}}$ in the transition diagram of hexangulations.