Atsuhiro Nakamoto

Note on irreducible triangulations of surfaces

In this paper, we shall show that an irreducible triangulation of a closed surface F2 has at most cg vertices, where g stands for a genus of F2 and c a constant.

Diagonal transformations in quadrangulations of surfaces

In this paper, it will be shown that any two bipartite quadrangulations of any closed surface are transformed into each other by two kinds of transformations, called the diagonal slide and the diagonal rotation, up to homeomorphism, if they have the same and sufficiently large number of vertices.

Diagonal Transformations and Cycle Parities of Quadrangulations on Surfaces

In this paper, we shall show that any two quadrangulations on any closed surface can be transformed into each other by diagonal slides and diagonal rotations if they have the same and sufficiently large number of vertices and if the homological properties of both quadrangulations coincide.

Irreducible Quadrangulations of the Torus

In 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.

Generating quadrangulations of surfaces with minimum degree at least 3

A graph G is said to be Pt-free if it does not contain an induced path on t vertices. The i-center Ci(G) of a connected graph G is the set of vertices whose distance from any vertex in G is at most i. Denote by I(t) the set of natural numbers i, ⌊t-2⌋ ≤ i ≤ t - 2, with the property that, in every connected Pt-free graph G, the i-center Ci(G) of G induces a connected subgraph of G. In this article, the sharp upper bound on the diameter of G[Ci(G)] is established for every i ∈ I(t). The sharp lower bound on I(t) is obtained consequently.

Diagonal Flips in Triangulations on Closed Surfaces with Minimum Degree at Least 4

It will be shown that any two triangulations on a closed surface, except the sphere, with minimum degree at least 4 can be transformed into each other by a finite sequence of diagonal flips through those triangulations if they have a sufficiently large and same number of vertices. The same fact holds for the sphere if they are not equivalent to a double wheelCn+K2.

Diagonal Flips in Hamiltonian Triangulations on the Sphere

Abstract.In this paper, we shall prove that any two Hamiltonian triangulations on the sphere with n≥5 vertices can be transformed into each other by at most 4n−20 diagonal flips, preserving the existence of Hamilton cycles. Moreover, using this result, we shall prove that at most 6n−30 diagonal flips are needed for any two triangulations on the sphere with n vertices to transform into each other.

Fractional chromatic numbers of cones over graphs

It has been shown that every quadrangulation on any nonspherical orientable closed surface with a suf®ciently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface Nk has chromatic number at least 4 if G has a cycle of odd length which cuts open Nk into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface Nk admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity. ß 2001 John Wiley & Sons, Inc. J Graph Theory 37: 100±114, 2001

Diagonal Flips of Triangulations on Closed Surfaces Preserving Specified Properties

Consider 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.”

Subgraphs of graphs on surfaces with high representativity

Let G be a 3-connected graph with n vertices on a non-spherical closed surface Fk2 of Euler genus k with sufficiently large representativity. In this paper, we first study a new cutting method which produces a spanning planar subgraph of G with a certain good property. This is used to show that such a graph G has a spanning 4-tree with at most max{2k-5,0} vertices of degree 4. Using this result, we prove that for any integer t, if n is sufficiently large, then G has a connected subgraph with t vertices whose degree sum is at most 8t-1. We also give a nearly sharp bound for the projective plane, torus and Klein bottle. Furthermore, using our cutting method, we prove that a 3-connected graph G on Fk2 with high representativity has a 3-walk in which at most max{2k-4,0} vertices are visited three times, and an 8-covering with at most max{4k-8,0} vertices of degree 7 or 8. Moreover, a 4-connected G has a 4-covering with at most max{4k-6,0} vertices of degree 4.