Transformations in hexangulations on the sphere

A hexangulation G is a plane graph with each face bounded by a cycle of length six. Define three kinds of transformations A, B and C for hexangulations. It was proved that any two hexangulations with the same order can be transformed into each other by a repeated application of A, B and C, and that without any one of the three transformations, there is a hexangulation which cannot be transformed only by the remaining two transformations. s article, focusing on the fact that a hexangulation is always bipartite, we consider transformations of hexangulations preserving its bipartition. Let G be any hexangulation with bipartition V ( G ) = B ∪ W , where | B | ⩾ | W | , and we call ( | B | , | W | ) the bipartition size of G. Observe that both A and C preserve the bipartition size, but B does not. Then we first prove that G satisfies | B | ⩽ 3 | W | − 6 . Secondly, for any hexangulation G ′ with the same partition size of G, we prove that if | B | < 3 | W | − 6 , then G can be transformed into G ′ only by A and C, and if | B | = 3 | W | − 6 , then G can be transformed into G ′ only by B 2 , where B 2 is a transformation preserving the bipartition size and consisting of B twice.