Stefan Felsner
We show that 4-connected plane triangulations can be redrawn such that edges are represented by straight segments and the vertices are covered by a set of at most $\sqrt{2n}$ lines each of them horizontal or vertical. The same holds for all subgraphs of such triangulations. The proof is based on a corresponding result for diagrams of planar lattices which makes use of orthogonal chain and antichain families.
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We prove that 3-connected triangulations with at most one separating triangle are hamiltonian-connected. In order to show bounds on the strongest form of this theorem, we proved that for any $s\geq4$ there are 3-connected triangulation with $s$ separating triangles that are not hamiltonian-connected. We also present computational results which show that all `small' 3-connected triangulations with at most 3 separating triangles are hamiltonian-connected.
A model of trees for 5-connected planar triangulations
本研究建立了5c-三角剖分与满足内边局部约束的5-正则平面树的双射关系，推导了相关生成函数的显式表达式，并为5-连通平面三角剖分的高效均匀随机生成和简洁编码提供了可行途径。
Whitney proved in 1931 that every 4-connected planar triangulation is hamiltonian. Later in 1979, Hakimi, Schmeichel and Thomassen conjectured that every such triangulation on $n$ vertices has at least $2(n - 2)(n - 4)$ hamiltonian cycles. Along this direction, Brinkmann, Souffriau and Van Cleemput established a linear lower bound on the number of hamiltonian cycles in 4-connected planar triangulations. In stark contrast, Alahmadi, Aldred and Thomassen showed that every 5-connected triangulation of the plane or the projective plane has exponentially many hamiltonian cycles. This gives the moti…
A $3$-connected graph $G$ is essentially $4$-connected if, for any $3$-cut $S\subseteq V(G)$ of $G$, at most one component of $G-S$ contains at least two vertices. We prove that every essentially $4$-connected maximal planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2}{3}(n+4)$; moreover, this bound is sharp.
On the number of 4-contractible edges in plane triangulations
本文改进了Ando和Egawa关于4连通图中保持4连通性的可缩边下界定理，特别是针对4连通的平面三角剖分，证明了当图阶数至少为7时，至少包含|V_{≥5}|+2个保持4连通性的可缩边，并确定了极值图。
Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we show that every $4$-connected planar triangulation on $n$ vertices has $Ω(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{Ω(n^{1/4})}$ Hamiltonian cycles.
The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide formulae for the minimum Wiener index of simple triangulations and quadrangulations with connectivity at least $c$, and provide the extremal structures, which attain those values. Our main tool is setting upper bounds for the maximum degree in highly connected triangulations and quadrangulations.
Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices then $G$ contains at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar $n$-vertex triangulations $G$ that do not have too many separating 4-cyc…
An ear in a triangulation $T$ of a convex $n$-gon $P$ is a triangle of $T$ that shares two sides with $P$ itself. Certain enumerational and structural problems become easier when one considers only triangulations with few ears. We demonstrate this in two ways. First, for $k=2, 3$, we find the number of symmetry classes of triangulations with $k$ ears. Second, for $k=2, 3$, we determine the number of triangulations disjoint from a given triangulation: this number depends only on $n$ for $k=2$, and only on lengths of branches of the dual tree for $k=3$.
Counting $k$-cycles in $5$-connected planar triangulations
该研究证明了5-连通平面三角剖分中5圈数上界为9n-50且紧，并给出k≥6时k圈数的渐近紧上界C(k)·n^{⌊k/3⌋}。
An embedded graph is called $z$-knotted if it contains the unique zigzag (up to reversing). We consider $z$-knotted triangulations, i.e. $z$-knotted embedded graphs whose faces are triangles, and describe all cases when the connected sum of two $z$-knotted triangulations is $z$-knotted.
A small triangulation of the sphere product can be found in lower dimensions by computer search and is known in few other cases: Klee and Novik constructed a centrally symmetric triangulation of $\mathbb{S}^i\times \mathbb{S}^{d-i-1}$ with $2d+2$ vertices for all $d\geq 3$ and $1\leq i\leq d-2$; they also proposed a balanced triangulation of $\mathbb{S}^1\times \mathbb{S}^{d-2}$ with $3d$ or $3d+2$ vertices. In this paper, we provide another centrally symmetric $(2d+2)$-vertex triangulation of $\mathbb{S}^2\times \mathbb{S}^{d-3}$. We also construct the first balanced triangulation of $\mathbb…
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…
Transversal structures (also known as regular edge labelings) are combinatorial structures defined over 4-connected plane triangulations with quadrangular outer-face. They have been intensively studied and used for many applications (drawing algorithm, random generation, enumeration ...). In this paper we introduce and study a generalization of these objects for the toroidal case. Contrary to what happens in the plane, the set of toroidal transversal structures of a given toroidal triangulation is partitioned into several distributive lattices. We exhibit a subset of toroidal transversal struc…
In this paper, we consider the problem of determining in polynomial time whether a given planar point set $P$ of $n$ points admits 4-connected triangulation. We propose a necessary and sufficient condition for recognizing $P$, and present an $O(n^3)$ algorithm of constructing a 4-connected triangulation of $P$. Thus, our algorithm solves a longstanding open problem in computational geometry and geometric graph theory. We also provide a simple method for constructing a noncomplex triangulation of $P$ which requires $O(n^2)$ steps. This method provides a new insight to the structure of 4-connect…
Cyclically Colored Triangulations: Enumeration and Connectedness of Reconfiguration Graphs
该研究针对循环着色凸多边形有效三角剖分的重构图，通过根边分解方法分别解决了3色情形的连通性猜想与计数递推，证明了4色及以上翻转图的连通性，建立了统一的结构分析框架。
Let $Γ$ be a triangulation of a connected closed $2$-dimensional (not necessarily orientable) surface. Using zigzags (closed left-right paths), for every face of $Γ$ we define the $z$-monodromy which acts on the oriented edges of this face. There are precisely $7$ types of $z$-monodromies. We consider the following two cases: (M1) the $z$-monodromy is identity, (M2) the $z$-monodromy is the consecutive passing of the oriented edges. Our main result is the following: the subgraphs of the dual graph $Γ^{*}$ formed by edges whose $z$-monodromies are of types (M1) and (M2), respectively, both are …
Refining a classical proof of Whitney, we show that any $4$-connected planar triangulation can be decomposed into a Hamiltonian path and two trees. Therefore, every $4$-connected planar graph decomposes into three forests, one having maximum degree at most $2$. We use this result to show that any Hamiltonian planar triangulation can be decomposed into two trees and one spanning tree of maximum degree at most $3$. These decompositions improve the result of Gonçalves [Covering planar graphs with forests, one having bounded maximum degree. J. Comb. Theory, Ser. B, 100(6):729--739, 2010] that ever…
A Schnyder-type drawing algorithm for 5-connected triangulations
本文为5-连通三角剖分定义Schnyder型组合结构，并证明基于重心放置的绘图算法可生成平面直线图。
In this paper, we show that two balanced triangulations of a closed surface are not necessary connected by a sequence of balanced stellar subdivisions and welds. This answers a question posed by Izmestiev, Klee and Novik. We also show that two balanced triangulations of a closed surface are connected by a sequence of three local operations, which we call the pentagon contraction, the balanced edge subdivision and the balanced edge weld. In addition, we prove that two balanced triangulations of the 2-sphere are connected by a sequence of pentagon contractions and their inverses if none of them …
In this paper we prove that a finite triangulation of a connected closed surface is completely determined by its intersection matrix. The \emph{intersection matrix} of a finite triangulation, $K$, is defined as $M_{K}=(dim(s_{i}\cap s_{j}))_{0\leq i,0\leq j}^{n-1}$, where $K_{2}=\{s_{0}, \ldots s_{n-1}\}$ is a labelling of the triangles of $K$.
Wiener Index and Remoteness in Triangulations and Quadrangulations
本文研究简单三角剖分和四边形剖分中Wiener指数的渐近最大值及遥远度的尖锐上界，并基于计算证据提出极值结构的猜想。
We give a short proof of a result of Jordan and Tanigawa that a 4-connected graph which has a spanning planar triangulation as a proper subgraph is generically globally rigid in R^3. Our proof is based on a new sufficient condition for the so called vertex splitting operation to preserve generic global rigidity in R^d.
A zigzag in a map (a $2$-cell embedding of a connected graph in a connected closed $2$-dimensional surface) is a cyclic sequence of edges satisfying the following conditions: 1) any two consecutive edges lie on the same face and have a common vertex, 2) for any three consecutive edges the first and the third edges are disjoint and the face containing the first and the second edges is distinct from the face which contains the second and the third. A map is $z$-knotted if it contains a single zigzag. Such maps are closely connected to Gauss code problem and have nice homological properties. We s…
We show that no minimal vertex triangulation of a closed, connected, orientable 2-manifold of genus 6 admits a polyhedral embedding in R^3. We also provide examples of minimal vertex triangulations of closed, connected, orientable 2-manifolds of genus 5 that do not admit any polyhedral embeddings. We construct a new infinite family of non-realizable triangulations of surfaces. These results were achieved by transforming the problem of finding suitable oriented matroids into a satisfiability problem. This method can be applied to other geometric realizability problems, e.g. for face lattices of…
Connected Domination in Plane Triangulations
本文研究平面三角剖分的连通支配数，计算至多13个顶点的全部情形，构造满足模3条件的极值图，并证明连通支配数与支配数之差可任意大。
On the oriented diameter of near planar triangulations
本文研究近平面三角剖分的有向直径，证明除七个小例外外，$n$ 顶点 $2$-连通近三角剖分的有向直径不超过 $\lceil n/2\rceil$ 且界紧，改进并推广了已有结果。
The enumeration of plane (3,6)-triangulations and the form of odd perfect numbers
该研究计数了顶点度数为3或6的3-连通平面三角剖分族中特定度数的非同构剖分数目，并借助奇数参数下的计数公式强化了关于奇完全数形式的欧拉定理。
Connected Dominating Sets in Triangulations
本文证明任意n顶点三角剖分存在大小至多10n/21的连通支配集，等价于存在至少11n/21个叶子的生成树，改进了此前n/2的界，并应用于SEFENOMAP图绘制问题及亏格g曲面三角剖分。
Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation is a full triangulation of some subset P' of P containing all extreme points in P. A bistellar flip on a partial triangulation flips an edge (an edge flip), removes a non-extreme point of degree 3, or adds a point in P \ P' as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The edge flip…
We describe an algorithm for the enumeration of (candidates of) vertex-transitive combinatorial $d$-manifolds. With an implementation of our algorithm, we determine, up to combinatorial equivalence, all combinatorial manifolds with a vertex-transitive automorphism group on $n\leq 13$ vertices. With the exception of actions of groups of small order, the enumeration is extended to 14 and 15 vertices.
In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances. Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit…
In 1996, Tarjan and Matheson proved that if $G$ is a plane triangulated disc with $n$ vertices, $γ(G)\le n/3$, where $γ(G)$ denotes the domination number of $G$. Furthermore, they conjectured that the constant $1/3$ could be improved to $1/4$ for sufficiently large $n$. Their conjecture remains unsettled. In the present paper, it is proved that if $G$ is a hamiltonian plane triangulation with $|V(G)|=n$ vertices and minimum degree at least 4, then $γ(G)\le\max\{\lceil 2n/7\rceil, \lfloor 5n/16\rfloor\}$. It follows immediately that if $G$ is a 4-connected plane triangulation with $n$ vertices,…
A centrally symmetric $2d$-vertex combinatorial triangulation of the product of spheres $§^i\times§^{d-2-i}$ is constructed for all pairs of non-negative integers $i$ and $d$ with $0\leq i \leq d-2$. For the case of $i=d-2-i$, the existence of such a triangulation was conjectured by Sparla. The constructed complex admits a vertex-transitive action by a group of order $4d$. The crux of this construction is a definition of a certain full-dimensional subcomplex, $\B(i,d)$, of the boundary complex of the $d$-dimensional cross-polytope. This complex $\B(i,d)$ is a combinatorial manifold with bounda…
Spanning trees for many different numbers of leaves
本文证明连通图的生成树叶子数集合具有连续性，并结合 Whitney 定理给出平面4-连通三角剖分中可实现叶子数范围的构造性结果，同时证明该范围的最优性。
A plane near-triangulation G can be decomposed into a collection of induced subgraphs, described here as the W-components of G, such that G is perfect (respectively, chordal) if and only if each of its W-components is perfect (respectively, chordal). Each W-component is a 2-connected plane near-triangulation, free of edge separators and separating triangles. Graphs satisfying these conditions will be called W-near-triangulations. A linear time decomposition of G into its W-components is achievable using known techniques from the literature. W-near-triangulations have the property that the open…
The Stein theorem for loopless 2-connected plane multigraphs
本文将Stein定理从简单平面三角剖分推广到无环2-连通平面多重图，并给出了若干等价刻画。
Disconnected cuts in 4-connected planar graphs
本文研究4-连通平面图的极小割集，证明其所有极小割集连通当且仅当该图为近三角剖分，并基于此给出寻找不连通割集的线性时间算法框架。
The problem of finding a triangulation of a convex three-dimensional polytope with few tetrahedra is proved to be NP-hard. We discuss other related complexity results.
Counting symmetric unimodular triangulations
研究平面扩张标准三角的三角剖分，聚焦具有固定对称轴的单模三角剖分，给出上下界并枚举小案例。
The main objects of the paper are $z$-oriented triangulations of connected closed $2$-dimensional surfaces. A $z$-orientation of a map is a minimal collection of zigzags which double covers the set of edges. We have two possibilities for an edge -- zigzags from the $z$-orientation pass through this edge in different directions (type I) or in the same direction (type II). Then there are two types of faces in a triangulation: the first type is when two edges of the face are of type I and one edge is of type II and the second type is when all edges of the face are of type II. We investigate $z$-o…
This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant b_M for simplicial d-manifolds M taking values in the range 0 <= b_M <= d-1. The main result is that b_M influences connectivity in the following way: The graph of a d-dimensional simplicial compact manifold M is (2d - b_M)-connected. The parameter b_M has the property that b_M = 0 if the complex M is flag. Hence, our result interpolates between Barnette's theorem (1982) that all d-manifold graphs are (d+1)-connected and …
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。
Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every $n$-vertex triangulation with at least six vertices has a simultaneous flip into a 4-connected triangulation, and that it can be computed in O(n) time. It follows that every triangulation has a simultaneous flip into a Hamiltonian triangulation. This result is used to prove that for any two $n$-vertex triangulations, there exists a sequence of $O(\log n)$ simultaneous flips to transform one into the other. The total number of edges flipped in this sequence is O(n). The maximum size of a simultaneous f…
We construct small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected. These examples improve our construction in J. Amer. Math. Soc., 13:3 (2000), 611--637 not only in size, but also in that their toric Hilbert schemes are not connected either, a question left open in that article. Additionally, the point sets can easily be put into convex position, providing examples of 5-dimensional polytopes with non-connected graph of triangulations.
Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our labeling resembles many of the properties of Schnyder's one for triangulations: Apart from being in bijection with tree decompositions, paths in these trees allow to define the regions of a vertex such that counting faces in them yields an algorithm for embedding the quadrangulation, in this case on a~2-book. Furthermore, as Schnyder labelings have been extended t…
We prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension $d \geq 4$, if $Δ$ is a tight connected closed homology $d$-manifold whose $i$th homology vanishes for $1 < i < d-1$, then $Δ$ is a stacked triangulation of a manifold.These results give affirmative answers to questions posed by Novik and Swartz and by Effenberger.
This paper deals with triangulations of the 2-torus with the vertex labeled general octahedral graph $O_4$ which is isomorphic to the complete four-partite graph $K_{2,2,2,2}$; it is known that there exist precisely twelve such triangulations. We find all the 12 triangulations in a Schlegel diagram of the hyperoctahedron and realize all of them geometrically with the same 1-skeleton in 3-space. In particular, we identify two geometric polyhedral tori (both without self-intersections) with the same 1-skeleton in 3-space, but without a single common face, or in other words their intersection (as…
Non-Hamiltonian $\frac{3}{2}$-Tough Plane Triangulations
该研究否定了Hao等人提出的‘度数为3的顶点两两距离至少3的3/2-坚韧平面三角剖分是哈密顿图’的问题，证明了存在满足任意长距离条件的此类非哈密顿图，明确哈密顿圈的障碍源于局部构型而非其邻近性。
A non-constructive proof of the Four Colour Theorem
本文利用生成函数奇异性分析、渐近分析、超几何级数性质及Tutte的平面地图与色多项式枚举工作，给出四色定理的非构造性证明。
We explicitly construct small triangulations for a number of well-known 3-dimensional manifolds and give a brief outline of some aspects of the underlying theory of 3-manifolds and its historical development.
3-packings in Triangulations: Algorithms, bounds, and Complexity
研究平面三角剖分中三种三顶点图H的H-打包问题，给出P3-打包的下界并证明渐近紧性，利用哈密顿圈刻画4-连通三角剖分中的三角形因子，并证明诱导P2∪P1-打包的多项式时间可解性。
We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) $3$-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the $3$-connected case) to this setting. Precisely, for any cylindric essentially internally $3$-connected map $G$ with $n$ vertices, we can obtain in linear time a periodic (in $x$) straight-line drawing …
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