Dominating Sets in Plane Triangulations
Erika L. C. King, Michael J. Pelsmajer
在1996年，Matheson和Tarjan证明了任何n-顶点三角剖分，如果n足够大，则其控制集的大小不超过n/4。我们证明了这一点的最大度6图。
In 1996, Matheson and Tarjan conjectured that any n-vertex triangulation with n sufficiently large has a dominating set of size at most n/4. We prove this for graphs of maximum degree 6.
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Dominating Sets in Triangulations on Surfaces
图G的一个控制集D是一个集合，使得G的每个顶点v要么在该集合中，要么与该集合中的一个顶点相邻。Matheson和Tarjan（1996）证明了任意n-顶点平面三角剖分都有一个大小不超过n/3的控制集，并且当n足够大时，给出了n/4的界。King和Pelsmajer最近证明了这一点，最大度至多为6。Plummer和Za（2009）以及Honjo、Kawarabayashi和Nakamoto（2009）将n/3界扩展到曲面上的三角剖分。我们证明了两个相关的结果：（i）存在一个常数c，使得任何最大度不超过6的n顶点平面三角剖分都有一个大小不超过n/6 + c的控制集。(ii)对于任意曲面S，非负t，且n> 0，存在C使得对于S上任意n-顶点三角剖分且至多t个顶点的度数不为6，存在大小至多为n（1/6 + n）+ C的控制集。作为证明的一部分，我们还证明了任何一个非可定向曲面的n-顶点三角剖分有一个长度不超过2sqrt（n）的不可收缩圈。Albertson和Hutchinson（1986）证明了对于非球面的可定向曲面的n-顶点三角剖分有一个长度为sqrt（2n）的不可收缩圈，但对于非可定向曲面还没有类似的结果。
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].
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,…
Let $G$ be a graph and $γ(G)$ denote the domination number of $G$, i.e. the cardinality of a smallest set of vertices $S$ such that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. Matheson and Tarjan conjectured that a plane triangulation with a sufficiently large number $n$ of vertices has $γ(G)\le n/4$. Their conjecture remains unsettled. In the present paper, we show that: (1) a maximal outerplane graph with $n$ vertices has $γ(G)\le \lceil \frac{n+k} 4\rceil$ where $k$ is the number of pairs of consecutive degree 2 vertices separated by distance at least 3 on the bound…
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。
We introduce a class of plane graphs called weak near-triangulations, and prove that this class is closed under certain graph operations. Then we use the properties of weak near-triangulations to prove that every plane triangulation on $n>6$ vertices has a dominating set of size at most $17n/53$. This improves the bound $n/3$ obtained by Matheson and Tarjan.
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⌋。
Triangulations Admit Dominating Sets of Size $2n/7$
本文证明了平面三角剖分存在大小为$2n/7$的支配集，逼近Matheson和Tarjan的猜想，通过引入惩罚函数与骨架三角剖分类进行归纳证明，并给出了二次时间的构造算法。
Maximal Independent Sets in Planar Triangulations
本研究证明含n个顶点的平面三角剖分存在大小不超n/3的极大独立集，证实了相关猜想，是对三角化圆盘控制集经典结论的结构性强化。
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。
Face-hitting Dominating Sets in Planar Graphs
本文证明无孤立点、自环和2面的平面多重图可二分为两个支配且面命中的顶点集，并据此改进特定平面三角剖分的支配集上界。
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。
We characterise $t$-perfect plane triangulations by forbidden induced subgraphs. As a consequence, we obtain that a plane triangulation is $h$-perfect if and only if it is perfect.
We prove that a triangulation of the projective plane is (strongly) $t$-perfect if and only if it is perfect and contains no $K_4$.
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 $\gamma_t(G)\le\lfloor2n/5\rfloor$。
Spanning bipartite quadrangulations of triangulations of the projective plane
本文研究了射影平面三角形的生成双色四边形化子图特性，解决了Kündgen和Ramamurthi在2002年提出的问题。
We describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result which shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a dominating set in a projective plane of order $q>81$ is smaller than $2q+2[\sqrt{q}]+2$ (i.e., twice the size of a Baer subplane), then it contains either all but possibly one points of a line or all but possibly one lines through a point. Furthermore, we completely characterize dominating sets of size at most $2q+\sqrt{q}+1$. In Desarguesian planes, we could…
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。
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。
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…
Contributions to conjectures on planar graphs: Induced Subgraphs, Treewidth, and Dominating Sets
本文澄清了平面图中连通支配与导出外平面子图等概念的关系，构造了连通支配数超过 n/3 的平面三角剖分无限族，否定了 Bradshaw 等人的问题，并获得了有界树宽导出子图的新结果。
A 2-tree is a graph that can be formed by starting with a triangle and iterating the operation of making a new vertex adjacent to two adjacent vertices of the existing graph. Leizhen Cai asked in 1995 whether every maximal planar graph contains a spanning 2-tree. We answer this question in the negative by constructing an infinite class of maximal planar graphs that have no spanning 2-tree.
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 $2n+3t-1…
Null and non--rainbow colorings of projective plane and sphere triangulations
对于$n\geq 4$阶的极大可平面图，我们证明了不含彩虹面的顶点着色最多使用$\lfloor\frac{2n-1}{3}\rfloor$颜色，这是最好的可能。对于嵌入在射影平面上的极大图，我们得到了类似的最佳界$\lfloor\frac{2n+1}{3}\rfloor$。在证明的主要成分是经典的同调工具。通过将图看作拓扑空间，引入了零染色的概念，证明了对任何图$G$，极大零染色$f$使得商图$G/f$是森林.
We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions.
For integers $n\ge 0$, an iterated triangulation $Tr(n)$ is defined recursively as follows: $Tr(0)$ is the plane triangulation on three vertices and, for $n\ge 1$, $Tr(n)$ is the plane triangulation obtained from the plane triangulation $Tr(n-1)$ by, for each inner face $F$ of $Tr(n-1)$, adding inside $F$ a new vertex and three edges joining this new vertex to the three vertices incident with $F$. In this paper, we show that there exists a 2-edge-coloring of $Tr(n)$ such that $Tr(n)$ contains no monochromatic copy of the cycle $C_k$ for any $k\ge 5$. As a consequence, the answer to one of two …
For a set $A$ of points in the plane, not all collinear, we denote by ${\rm tr}(A)$ the number of triangles in any triangulation of $A$; that is, ${\rm tr}(A) = 2i+b-2$ where $b$ and $i$ are the numbers of points of $A$ in the boundary and the interior of $[A]$ (we use $[A]$ to denote "convex hull of $A$"). We conjecture the following analogue of the Brunn-Minkowski inequality: for any two point sets $A,B \subset {\mathbb R}^2$ one has \[ {\rm tr}(A+B)^{\frac12}\geq {\rm tr}(A)^{\frac12}+{\rm tr}(B)^{\frac12}. \] We prove this conjecture in several cases: if $[A]=[B]$, if $B=A\cup\{b\}$, if $|…
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…
A characterisation of 3-colourable 3-dimensional triangulations
将Heawood定理推广至三维三角剖分，证明边三色可着色等价于所有边度数为偶数。
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.
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…
We prove that any triangulation of a surface different from the sphere and the projective plane admits an orientation without sinks such that every vertex has outdegree divisible by three. This confirms a conjecture of Barát and Thomassen and is a step towards a generalization of Schnyder woods to higher genus surfaces.
We derive a local criterion for a plane near-triangulated graph to be perfect. It is shown that a plane near-triangulated graph is perfect if and only if it does not contain either a vertex, an edge or a triangle, the neighbourhood of which has an odd hole as its boundary. The characterization leads to an $O(n^2)$ algorithm for checking perfectness of plane near-triangulations.
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 better upper bound on the number of triangulations of a planar point set
证明了平面上基数为$n$的点集不能是其凸包的多个$59^n O(n^{-6})$直边三角剖分的顶点集。这改进了$276.75^n$的先前上界。
Simultaneous Diagonal Flips in Plane Triangulations
研究了平面三角剖分中的同时对角翻转问题。证明了每个至少有6个顶点的$n$ -顶点三角剖分都有一个同时翻转成4-连通三角剖分，并且可以在O（n）时间内计算.因此，每个三角剖分都同时翻转成一个哈密尔顿三角剖分。这个结果被用来证明，对于任何两个$n$ -顶点三角剖分，存在一个序列的$O(\log n)$同时翻转转换成其他。因此，时间复杂度为O（n）。同时翻转的最大尺寸，然后研究。证明了每个三角剖分至少有${1/3}(n-2)$边同时翻转。另一方面，每个同时翻转最多具有$n-2$边，并且存在具有最大同时翻转的${6/7}(n-2)$边的三角剖分。
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.
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…
A lozenge triangulation of the plane with integers
本文引入三参数三重线性算子，将整数三元组分解为四个二维三角剖分塔，并完整刻画各网络中的整数为二次形式族，其中一塔由胚生成Löschian数覆盖平面。
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…
Plane perfect matchings of $2n$ points in convex position are in bijection with triangulations of convex polygons of size $n+2$. Edge flips are a classic operation to perform local changes both structures have in common. In this work, we use the explicit bijection from Aichholzer et al. (2018) to determine the effect of an edge flip on the one side of the bijection to the other side, that is, we show how the two different types of edge flips are related. Moreover, we give an algebraic interpretation of the flip graph of triangulations in terms of elements of the corresponding Temperley-Lieb al…
Orbits and Hamilton bonds in a family of plane triangulations with vertices of degree three or six
设$\cal{P}$是所有顶点为三次或六次的2-连通平面三角剖分的族。Grünbaum和Motzkin证明了（在对偶项中）每个图$P \in \cal{P}$都可以分解为因子$P_0$、$P_1$、$P_2$（由循环群$Q = \{0,1,2\}$的元素索引），使得每个因子$P_q$都由两个长度相同的诱导路径$M(q)$和长度相同的$K(q)-1$诱导循环组成。对于$q \in Q$，我们定义一个整数$S^+(q)$，使得向量$(K(q), M(q), S^+(q))$唯一地确定图$P$（如果$P$是简单的），直到方向保持同构。我们建立算术方程，该方程允许通过矢量$(K(q), M(q), S^+(q))$、$q \in Q$计算矢量$(K(q+1), M(q+1), S^+(q+1))$。我们提出了一些应用的方程。集合$\{(K(q), M(q), S^+(q)): q \in Q\}$称为$P$的轨道。我们刻画了$\cal{P}$中图的一点轨道。证明了若$P$的阶为$4n +2$，$n \in\mathbb{N}$，则它有一个Hamilton键，使得键的端树是均匀2-可着色的，且具有相同的阶.证明了如果$M(q)$是奇数且$K(q) \geqslant \frac{M(q)}{3}$是奇数，则存在两条不相交的同阶诱导路，它们的顶点一起跨越整个$P$.
A Quantitative Steinitz Theorem for Plane Triangulations
我们给出了一个新的证明斯坦尼茨的经典定理的情况下，平面三角剖分，这使我们能够获得一个新的一般性的约束上的网格大小的单纯多面体实现一个给定的三角剖分，次指数在一些特殊情况下。形式上，我们证明了每一个平面三角剖分$G$与$n$顶点可以嵌入在$\mathbb{R}^2$在这样一种方式，它是一个凸多面体表面的垂直投影。我们表明，该表面的顶点可以放置在一个$4n^3 \times 8n^5 \times ζ(n)$整数网格，其中$ζ(n) \leq (500 n^8)^{τ(G)}$和$τ(G)$表示的脱落直径的$G$，在该文件中定义的量。
There is no tame triangulation of the infinite real Grassmannian
我们证明了在R^ \infty中k平面的无限实格拉斯曼不存在三角剖分，它相对于坐标轴是很好的。在拟阵理论中，这表示没有格拉斯曼细分拟阵分层的三角剖分。这是证明了一个论点在射影几何，考虑一个特定的序列配置的点在平面上。
We prove that every triangulation of either of the torus, projective plane and Klein bottle, contains a vertex-spanning planar Laman graph as a subcomplex. Invoking a result of Kir{á}ly, we conclude that every $1$-skeleton of a triangulation of a surface of nonnegative Euler characteristic has a rigid realization in the plane using at most 26 locations for the vertices.
We study straight-line drawings of planar graphs with prescribed face areas. A plane graph is 'area-universal' if for every area assignment on the inner faces, there exists a straight-line drawing realizing the prescribed areas. For triangulations with a special vertex order, we present a sufficient criterion for area-universality that only requires the investigation of one area assignment. Moreover, if the sufficient criterion applies to one plane triangulation, then all embeddings of the underlying planar graph are also area-universal. Until now, it is open whether area-universality is a pro…
On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane
本文研究了八元数射影平面流形的27顶点三角剖分的对称群限制，利用Smith和Bredon的不动点集拓扑结果，确定了S_27的26个可能子群，其中最大群阶为351，其余群阶均不超过52。
Kempe equivalence of 4-colourings of some plane triangulations
本文研究了具有特定度数条件的平面三角剖分图$G_n$的4-染色问题，枚举了其Kempe类和不同4-染色的数量，并证明了去除极点后的子图$H_n$的所有4-染色在有限次Kempe变换内等价。
We prove that a planar graph is generically rigid in the plane if and only if it can be embedded as a pseudo-triangulation. This generalizes the main result of math.CO/0307347 which treats the minimally generically rigid case. The proof uses the concept of combinatorial pseudo-triangulation, CPT, in the plane and has two main steps: showing that a certain ``generalized Laman property'' is a necessary and sufficient condition for a CPT to be ``stretchable'', and showing that all generically rigid plane graphs admit a CPT assignment with that property. Additionally, we propose the study of combi…
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…
A construction of a $\frac{3}{2}$-tough plane triangulation with no 2-factor
本文通过构造一个没有2-因子的极大$\ rac{3}{2}$-坚韧平面图，填补了坚韧平面图中2-因子存在性问题的最后遗留情况，回答了Owens等人的疑问。
Binary Labelings for Plane Quadrangulations and their Relatives
出于双射平面三角剖分和分区的内部边缘成三棵树的施奈德标号之间，我们寻找二进制标签四边形（其边缘可以划分成两棵树）。我们的标签类似于许多的属性Schnyder的一个三角：除了在双射与树分解，在这些树中的路径允许定义的顶点的区域，这样计数面在他们产生一个算法嵌入四边形，在这种情况下，在一个~2-书。此外，由于Schnyder标号已经扩展到3-连通平面图，我们能够将我们的标号从四边形图扩展到更大的一类2-连通二部图。最后，我们提出了一个二元标号的拉曼图。
A 15-vertex triangulation of the quaternionic projective plane
本研究针对1992年Brehm与Kühnel构造的15顶点8维单纯复形是否为四元数射影平面最小三角剖分的悬而未决问题，通过实现Gaifullin的算法计算其第一庞特里亚金类，最终证实该复形确为四元数射影平面的最小三角剖分。
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。
Colouring the 1-skeleton of $d$-dimensional triangulations
本文在结构上刻画了${\mathbb S}^d$的三角剖分中具有$(d+2)$-可着色1-骨架的条件，填补了该领域20年来的研究空白。
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。
634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane
本文通过构造634个顶点传递及超过$10^{103}$个非顶点传递的27顶点组合16-流形，解决了类似于八元数射影平面的流形是否存在27顶点三角剖分的长期悬而未决的问题。
A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface $S$ is geometric, that is, it is combinatorially equivalent to a geodesic triangulation with respect to a constant curvature metric on $S$, and we list the possibilities. A key ingredient of the proof is to show that any two $d$-regular triangulations of the plane for $d> 6 $ are combinatorially equivalent. The proof of this uniqueness result, which is of independent interest, is based on an inductive argument involving some combina…
We give a general construction of triangulations starting from a walk in the quarter plane with small steps, which is a discrete version of the mating of trees. We use a special instance of this construction to give a bijection between maps equipped with a rooted spanning tree and walks in the quarter plane. We also show how the construction allows to recover several known bijections between such objects in a uniform way.
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。
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