Balanced subdivisions and flips on surfaces

Balanced triangulations on few vertices and an implementation of cross-flips

A $d$-dimensional simplicial complex is balanced if the underlying graph is $(d+1)$-colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and 3-manifolds on few vertices. In particular we construct small balanced triangulations of the 3-sphere that are non-shellable…

带长度或稀疏性约束的细分图的极值密度

Extremal density for subdivisions with length or sparsity constraints

本文研究图的平衡细分的极值密度问题，证明平均度的$e(H)$线性界保证任意图$H$的平衡细分存在，且该界在$H$对数稠密时最优，同时指出对数密度是临界阈值，次对数密度的满足可分性的几乎二部图可由次线性平均度界强制存在细分。

The $γ$-vector of a barycentric subdivision

We prove that the $γ$-vector of the barycentric subdivision of a simplicial sphere is the $f$-vector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used by Brenti and Welker to describe the $h$-vector of the barycentric subdivision of a boolean complex.

凸多边形的彩色三角剖分的翻转图

Flip graphs of coloured triangulations of convex polygons

本文研究凸多边形彩色三角剖分的翻转图，探讨其连通分量，并基于Gravier和Payan的结果，将四色定理与2-彩色三角剖分翻转图的连通性联系起来。

Geometric bistellar flips. The setting, the context and a construction

We give a self-contained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known results and connections to algebraic geometry, topological combinatorics, and other areas. As a new result, we announce the construction of a point set in general position with a disconnected space of triangulations. This shows, for the first time, that the poset of strict polyhedral subdivisions of a point set is not always connected.

Simplicial moves on balanced complexes

We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly $(d+1)$-colored) triangulation of a combinatorial $d$-manifold into another balanced triangulation. These moves form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following theorem: any two balanced triangulations of a closed combinatorial $d$-manifold can be connected by a sequence of cross-flips. Along the way we prove that for every $m \geq d+2$ and any closed combinatorial $d$-manifold $M$, two $m$-colored triangulations of $M$ can be connected by …

2-边距离平衡图

2-Edge Distance-Balanced Graphs

本文定义并研究2-边距离平衡图，提出利用偶环或奇环识别k=2,3的该类图的方法，并探讨笛卡尔积、字典序积及细分相关图的性质。

二维超三角剖分中的翻转

Flips in Two-dimensional Hypertriangulations

本文研究平面点集超三角剖分中的翻转，引入四种翻转类型，并证明水平-2超三角剖分通过这些翻转连通。

受限翻转图的S³三角剖分的树干

The Trunk of the Restricted Flip Graph of Triangulated S^3

研究了S³三角剖分的受限翻转图的结构，证明了树干在1-4翻转下的封闭性，并展示了其连通性和计算结果。

Balanced shellings and moves on balanced manifolds

A classical result by Pachner states that two $d$-dimensional combinatorial manifolds with boundary are PL homeomorphic if and only they can be connected by a sequence of shellings and inverse shellings. We prove that for balanced, i.e., properly $(d + 1)$-colored, manifolds such a sequence can be chosen such that balancedness is preserved in each step. As a key ingredient we establish that any two balanced PL homeomorphic combinatorial manifolds with the same boundary are connected by a sequence of basic cross-flips, as was shown recently by Izmestiev, Klee and Novik for balanced manifolds wi…

Geometric moves relate geometric triangulations

A geometric triangulation of a Riemannian manifold is a triangulation where the interior of each simplex is totally geodesic. Bistellar moves are local changes to the triangulation which are higher dimensional versions of the flip operation of triangulations in a plane. We show that geometric triangulations of a compact hyperbolic, spherical or Euclidean manifold are connected by geometric bistellar moves (possibly adding or removing vertices), after taking sufficiently many derived subdivisions. For dimensions 2 and 3, we show that geometric triangulations of such manifolds are directly relat…

Combinatorics of antiprism triangulations

The antiprism triangulation provides a natural way to subdivide a simplicial complex $Δ$, similar to barycentric subdivision, which appeared independently in combinatorial algebraic topology and computer science. It can be defined as the simplicial complex of chains of multi-pointed faces of $Δ$, from a combinatorial point of view, and by successively applying the antiprism construction, or balanced stellar subdivisions, on the faces of $Δ$, from a geometric point of view. This paper studies enumerative invariants associated to this triangulation, such as the transformation of the $h$-vector o…

Domino tilings of cylinders: the domino group and connected components under flips

We consider domino tilings of three-dimensional cubiculated regions. A flip is a local move: two neighboring parallel dominoes are removed and placed back in a different position. The twist is an integer associated to each tiling, which is invariant under flips. A balanced quadriculated disk $D$ is regular if whenever two tilings $t_0$ and $t_1$ of $D \times [0,N]$ have the same twist then $t_0$ and $t_1$ can be joined by a sequence of flips provided some extra vertical space is allowed. We define the domino group of a quadriculated disk and prove that $D$ is regular if and only if its domino …

Tilings of the sphere by congruent quadrilaterals I: edge combination $a^2bc$

The edge-to-edge tilings of the sphere by congruent quadrilaterals of Type $a^2bc$ are classified as $3$ classes: a sequence of two-parameter families of $2$-layer earth map tilings with $2n$ $(n\ge3)$ tiles, a one-parameter family of quadrilateral subdivisions of the octahedron with $24$ tiles together with a flip modification for a special parameter, and a sequence of $3$-layer earth map tilings with $8n$ $(n\ge2)$ tiles together with two flip modifications for odd $n$. We also describe the moduli and calculate the geometric data.