Irreducible Quadrangulations of the Torus

AbstractIn this paper, we find the irreducible quadrangulations of the torus. As a consequence, any two quadrangulations of the torus with the same number of vertices that are either both bipartite or both non-bipartite (except for some complete bipartite graphs) can be transformed into one another, up to homeomorphism, using a sequence of diagonal slides and diagonal rotations. We also determine the minor minimal 2-representative graphs on the torus

Acute triangles in triangulations on the plane with minimum degree at least 4

AbstractIn this paper, we show that every maximal plane graph with minimum degree at least 4 and m finite faces other than an octahedron can be drawn in the plane so that at least (m+3)/2 faces are acute triangles. Moreover, this bound is sharp

Diagonal flips in Hamiltonian triangulations on the projective plane

AbstractIn this paper, we shall prove that any two triangulations on the projective plane with n vertices can be transformed into each other by at most 8n-26 diagonal flips, up to isotopy. To prove it, we focus on triangulations on the projective plane with contractible Hamilton cycles

Diagonal Flips of Triangulations on Closed Surfaces Preserving Specified Properties

AbstractConsider a class P of triangulations on a closed surfaceF2, closed under vertex splitting. We shall show that any two triangulations with the same and sufficiently large number of vertices which belong to P can be transformed into each other, up to homeomorphism, by a finite sequence of diagonal flips through P. Moreover, if P is closed under homeomorphism, then the condition “up to homeomorphism” can be replaced with “up to isotopy.

Note On 6-regular Graphs On The Klein Bottle

Altshuler classified six regular graphs on the torus, but Thomassen and Negami gave different classifications for six regular graphs on the Klein bottle. In this note, we unify those two classifications, pointing out their difference and similarity

Quadrangulations and 2-Colorations

Any metric quadrangulation (made by segments of straight line) of a point set in the plane determines a 2-coloration of the set, such that edges of the quadrangulation can only join points with different colors. In this work we focus in 2-colorations and study whether they admit a quadrangulation or not, and whether, given two quadrangulations of the same 2-coloration, it is possible to carry one into the other using some local operations, called diagonal slides and diagonal rotation. Although the answer is negative in general, we can show a very wide family of 2-colorations, called onions 2-coloration, that are quadrangulable and which graph of quadrangulations is always connected

On separable self-complementary graphs

AbstractIn this paper, we describe the structure of separable self-complementary graphs

Diagonal flips in outer-triangulations on closed surfaces

We show that any two outer-triangulations on the same closed surface can be transformed into each other by a sequence of diagonal flips, up to isotopy, if they have a sufficiently large and equal number of vertices

Geometric Realization of Möbius Triangulations

A Möbius triangulation is a triangulation on the Möbius band. A geometric realization of a map M on a surface $\Sigma$ is an embedding of $\Sigma$ into a Euclidean 3-space $\mathbb{R}^3$ such that each face of M is a flat polygon. In this paper, we shall prove that every 5-connected triangulation on the Möbius band has a geometric realization. In order to prove it, we prove that if G is a 5-connected triangulation on the projective plane, then for any face f of G, the Möbius triangulation $G-f$ obtained from G by removing the interior of f has a geometric realization