Andrey Kupavskii, Aleksei Volostnov, Yury Yarovikov
We show that the number of partial triangulations of a set of $n$ points on the plane is at least the $(n-2)$-nd Catalan number. This is tight for convex $n$-gons. We also describe all the equality cases.
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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.
On the minimum number of non-monochromatic simplices for Sperner labelings of a regular triangulation
本文针对Mirzakhani和Vondrák提出的公开问题，研究了整数坐标顶点正则三角剖分中Sperner标号的非单色单纯形数量，并给出了其最小数的下界。
The problems of the genus of the complete graphs and minimum triangulations for each surface were both solved using the theory of current graphs, and each of them divided into twelve different cases, depending on the residue modulo 12 of the number of vertices. Cases 8 and 11 were of particular difficulty for both problems, with multiple families of current graphs developed to solve these cases. We solve these cases in a unified manner with families of current graphs applicable to both problems. Additionally, we give new constructions to both problems for Cases 6 and 9, which greatly simplify …
We study the minimum total weight of a disk triangulation using vertices out of $\{1,\ldots,n\}$, where the boundary is the triangle $(123)$ and the $\binom{n}3$ triangles have independent weights, e.g. $\mathrm{Exp}(1)$ or $\mathrm{U}(0,1)$. We show that for explicit constants $c_1,c_2>0$, this minimum is $c_1 \frac{\log n}{\sqrt n} + c_2 \frac{\log\log n}{\sqrt n} + \frac{Y_n}{\sqrt n}$ where the random variable $Y_n$ is tight, and it is attained by a triangulation that consists of $\frac14\log n + O_P(\sqrt{\log n}) $ vertices. Moreover, for disk triangulations that are canonical, in that n…
Random triangulations of the d-sphere with minimum volume
本文研究随机旅行商问题的高维推广，确定n个顶点上同胚于二维球面的最小体积随机三角剖分的增长速率与集中性，并对高维情形在现有知识范围内给出相应结果，同时提出与均匀随机三角剖分相联系的广义随机几何框架。
On weighted partial triangulations of convex polygons
本文研究了如何对凸多边形进行加权部分三角剖分的采样问题，提出了一种高效的精确采样方法，解决了传统马尔可夫链方法存在的混合时间不优问题。
Partial triangulations of surfaces with girth constraints
本文研究曲面上带围长约束的部分三角剖分与稀疏嵌入图，利用高亏格围长不等式刻画相关图类，并证明收缩极小的 (3,6)-紧或 (3,3)-紧 M-嵌入图具有有限性。
We determine the minimum number of vertices needed to provide balanced triangulations of $\mathbb S^{d-2}$-bundles over $\mathbb S^1$. If $d$ is odd and the bundle is orientable, or $d$ is even and the bundle is non-orientable, the minimum number of vertices is $3d$; otherwise, it is $3d+2$. Similar results apply to all balanced simplicial complexes that triangulate homology manifolds with $β_1\neq 0$ and $β_2=0$, where $β_i$'s are the Betti numbers, computed with coefficients in $\mathbb Q$.
We show that every simple planar near-triangulation with minimum degree at least three contains two disjoint total dominating sets. The class includes all simple planar triangulations other than the triangle. This affirms a conjecture of Goddard and Henning [Thoroughly dispersed colorings, J. Graph Theory, 88 (2018) 174-191].
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…
Independent dominating sets in planar triangulations
本文研究了平面三角剖分中独立支配集大小的最小上界问题，证明了近平面三角剖分的$ε$范围为$2/7 \leq ε\leq 5/12$，并在特定条件下进一步改进了上界。
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…
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,…
It is known that the flip distance between two triangulations of a convex polygon is related to the minimum number of tetrahedra in the triangulation of some polyhedron. It is interesting to know whether these two numbers are the same. In this work, we find examples to show that the two numbers are different in nature, and their ratio can be arbitrarily close to $\frac{3}{2}$.
Counting cycles in planar triangulations
该研究围绕平面三角剖分图的指定长度环计数问题，推导了不同环长度范围、不同类型平面三角剖分下环数的上下界，完善了平面图环计数领域的理论结果。
Paired and semipaired domination in triangulations
针对近三角剖分图，本文研究配对控制数与半配对控制数的上界，证明阶数n≥4时配对控制数不超过2⌊n/4⌋，n≥5时除少数例外半配对控制数不超过⌊2n/5⌋。
The triangulations of point configurations which decompose as a free sum are classified in terms of the triangulations of the summands. The methods employ two new kinds of partially ordered sets to be associated with any triangulation of a point set with one point marked, the "web of stars" and the "stabbing poset". Triangulations of smooth Fano polytopes are discussed as a case study.
We consider the so-called coupon-coloring of the vertices of a graph where every color appears in every open neighborhood, and our aim is to determine the maximal number of colors in such colorings. In other words, every color class must be a total dominating set in the graph and we study the total domatic number of the graph. We determine this parameter in every maximal outerplanar graph, and show that every Hamiltonian maximal planar graph has domatic number at least two, partially answering a conjecture of Goddard and Henning.
We compute the number of triangulations of a convex $k$-gon each of whose sides is subdivided by $r-1$ points. We find explicit formulas and generating functions, and we determine the asymptotic behaviour of these numbers as $k$ and/or $r$ tend to infinity. We connect these results with the question of finding the planar set of points in general position that has the minimum possible number of triangulations - a well-known open problem from computational geometry.
Given two graphs $G$ and $H$, the {\it rainbow number} $rb(G,H)$ for $H$ with respect to $G$ is defined as the minimum number $k$ such that any $k$-edge-coloring of $G$ contains a rainbow $H$, i.e., a copy of $H$, all of its edges have different colors. Denote by $M_t$ a matching of size $t$ and $\mathcal {T}_n$ the class of all plane triangulations of order $n$, respectively. Jendrol', Schiermeyer and Tu initiated to investigate the rainbow numbers for matchings in plane triangulations, and proved some bounds for the value of $rb({\mathcal {T}_n},M_t)$. Chen, Lan and Song proved that $2n+3t-1…
Given two graphs $G$ and $H$, the {\it rainbow number} $rb(G,H)$ for $H$ with respect to $G$ is defined as the minimum number $k$ such that any $k$-edge-coloring of $G$ contains a rainbow $H$, i.e., a copy of $H$, all of whose edges have different colors. Denote by $kK_2$ a matching of size $k$ and $\mathcal {T}_n$ the class of all plane triangulations of order $n$, respectively. In [S. Jendrol$'$, I. Schiermeyer and J. Tu, Rainbow numbers for matchings in plane triangulations, Discrete Math. 331(2014), 158--164], the authors determined the exact values of $rb(\mathcal {T}_n, kK_2)$ for $2\leq…
New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane
本文构造了大量四元数射影平面$\mathbb{HP}^2$的15顶点三角剖分新例子，并证明了在同构意义下恰好存在75个顶点数为15且对称群阶至少为4的$\mathbb{HP}^2$三角剖分。
A $triangulation$ is an embedding of a graph on surfaces where every face has length three. In this article, we show the existence of contractible Hamiltonian cycle in triangulated maps of which minimum degree is four.
Tuza (1981) conjectured that the size $τ(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $ν(G)$ of a maximum set of edge-disjoint triangles of $G$. In this paper we present three results regarding Tuza's Conjecture. We verify it for graphs with treewidth at most $6$; we show that $τ(G)\leq \frac{3}{2}\,ν(G)$ for every planar triangulation $G$ different from $K_4$; and that $τ(G)\leq\frac{9}{5}\,ν(G) + \frac{1}{5}$ if $G$ is a maximal graph with treewidth 3. Our first result strengthens a result of Tuza, implying that $τ(G) \leq 2\,ν(G)$ for…
On the number of perfect matchings in planar graphs
本文研究平面图中非零完美匹配的最小数量，证明在最小度为3的2连通平面图和3连通平面图中该数为常数，并给出无限族，最后构造反例反驳Zaks猜想。
Total domination in plane triangulations
本文研究近三角剖分的全支配数，证明除两个例外情形外，任意 $n\ge5$ 阶近三角剖分均满足 $\gamma_t(G)\le\lfloor2n/5\rfloor$。
Let $G$ be a triangulation of the sphere with vertex set $V$, such that the faces of the triangulation are properly coloured black and white. Motivated by applications in the theory of bitrades, Cavenagh and Wanless defined $A_W$ to be the abelian group generated by the set $V$, with relations $r+c+s=0$ for all white triangles with vertices $r$, $c$ and $s$. The group $A_B$ can be defined similarly, using black triangles. The paper shows that $A_W$ and $A_B$ are isomorphic, thus establishing the truth of a well-known conjecture of Cavenagh and Wanless. Connections are made between the structur…
The usual, or type A_n, Tamari lattice is a partial order on T_n^A, the triangulations of an (n+3)-gon. We define a partial order on T_n^B, the set of centrally symmetric triangulations of a (2n+2)-gon. We show that it is a lattice, and that it shares certain other nice properties of the A_n Tamari lattice, and therefore that it deserves to be considered the B_n Tamari lattice. We define a bijection between T_n^B and the non-crossing partitions of type B_n defined by Reiner. For S any subset of [n], Reiner defined a pseudo-type BD^S_n, to which is associated a subset of the noncrossing partiti…
Associahedra minimize $f$-vectors of secondary polytopes of planar point sets
本文推广了关于平点集三角剖分数量的下界结果，证明正则三角剖分已足够，并将结论扩展到所有正则细分，按次级多面体中面的维度分级。
On the minimum number of entries in a pair of maximal orthogonal partial Latin squares
研究证明了一对最大正交偏拉丁方中填充值的下限为$n^2/3$，并确定了该下限的唯一结构形式。
For a weighted outerplanar graph, the set of lex short cycles is known to be a minimum cycle basis [Inf. Process. Lett. 110 (2010) 970-974 ]. In this work, we show that the set of lex short cycles is a minimum cycle basis in weighted partial 2-trees (graphs of treewidth two) which is a superclass of outerplanar graphs.
The KP hierarchy is a completely integrable system of quadratic, partial differential equations that generalizes the KdV hierarchy. A linear combination of Schur functions is a solution to the KP hierarchy if and only if its coefficients satisfy the Plucker relations from geometry. We give a solution to the Plucker relations involving products of variables marking contents for a partition, and thus give a new proof of a content product solution to the KP hierarchy, previously given by Orlov and Shcherbin. In our main result, we specialize this content product solution to prove that the generat…
Superpolynomial lower bounds for vertex numbers of real projective space triangulations via a topological Figiel-Lindenstrauss-Milman theorem
本文通过拓扑Figiel-Lindenstrauss-Milman不等式证明实射影空间三角剖分顶点数的超多项式下界，并应用于图论和符号复形指标估计。
We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of Lyons and Peres. We prove that plane graphs of minimum degree at least $7$ have site percolation threshold bounded away from $1/2$, which was conjectured by Benjamini and Schramm, and make progress on a…
Partial cubes are graphs isometrically embeddable into hypercubes. We analyze how isometric cycles in partial cubes behave and derive that every partial cube of girth more than 6 must have vertices of degree less than 3. As a direct corollary we get that every regular partial cube of girth more than 6 is an even cycle. Along the way we prove that every partial cube $G$ with girth more than 6 is a tree-zone graph and therefore $2n(G)-m(G)-i(G)+ce(G)=2$ holds, where $i(G)$ is the isometric dimension of $G$ and $ce(G)$ its convex excess.
A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the problem of covering the cells of a line arrangement with a minimum number of lines, and the problem of finding a minimum-size fibre in a bipartite po…
A partial formula is provided to calculate the smallest number of vertices possible in a quadrangulation on the closed orientable 2-manifold of given genus. This extends the previously known partial formula due to N. Hartsfield and G. Ringel [J. Comb. Theory, Ser. B, 1989, 46, 84-95].
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