Werner Krauth, Martin Loebl
We expose a relationship between jamming and a generalization of Tutte's barycentric embedding. This provides a basis for the systematic treatment of jamming and maximal packing problems on two-dimensional surfaces.
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We present lower and upper bounds for the geometric-arithmetic index of unicyclic graphs and provide extremal graphs for the corresponding bounds.
A finite, simple and undirected graph $G = (V, E)$ with $p$ vertices and $q$ edges is said to be a $k$-geometric mean graph for a positive integer $k$ if there is an injection $ψ:V(G)\to \{k,k+1,\dots,k+q\}$ such that, when each edge $uv\in E(G)$ is assigned the label $\lfloor\sqrt{ψ(u)ψ(v)}\rfloor$ or $\lceil\sqrt{ψ(u)ψ(v)}\rceil$, the resulting edge label set is $\{k,k+1,...,k+q-1\}$ and $ψ$ is called a \emph{$k$-geometric mean labeling} of $G$. The special case $k=1$, a $1$-geometric mean labeling is called a geometric mean labeling, and a $1$-geometric mean graph is called a geometric mean…
The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index $GA_1$ and characterize graphs extremal with respect to them. Besides, we prove inequalities involving the geometric-arithmetic index of line graphs.
The game of cops and robbers is a pursuit game on graphs where a set of agents, called the cops try to get to the same position of another agent, called the robber. Cops and robbers has been studies on several classes of graphs including geometrically represented graphs. For example, it has been shown that string graphs, including geometric graphs, have cop number at most 15. On the other hand, little is known about geometric graphs of any cop number less than 15 and there is only one example of a geometric graph of cop number three that has as many as 1440 vertices. In this paper we present a…
For any metric $d$ on $\mathbb{R}^2$, an ($\mathbb{R}^2,d$)-geometric graph is a graph whose vertices are points in $\mathbb{R}^2$, and two vertices are adjacent if and only if their distance is at most 1. If $d=\|.\|_{\infty}$, the metric derived from the $L_{\infty}$ norm, then $(\mathbb{R} ^2,\|.\|_{\infty})$-geometric graphs are precisely those graphs that are the intersection of two unit interval graphs. We refer to $(\mathbb{R}^2,\|.\|_{\infty})$-geometric graphs as square geometric graphs. We represent a characterization of square geometric graphs. Using this characterization we provide…
We study Hamiltonicity in graphs obtained as the union of a deterministic $n$-vertex graph $H$ with linear degrees and a $d$-dimensional random geometric graph $G^d(n,r)$, for any $d\geq1$. We obtain an asymptotically optimal bound on the minimum $r$ for which a.a.s. $H\cup G^d(n,r)$ is Hamiltonian. Our proof provides a linear time algorithm to find a Hamilton cycle in such graphs.
A note on $χ$-unbounded classes of geometric graphs
本文证明当d≥3时存在无穷多个d-CBU几何相交图类不是χ-有界的，且与Burling图类不可比较，从而否定Burling图是几何相交类中无界色数唯一来源的猜想。
We introduce the notion of weighted Coxeter graph and associate to it a certain generalization of the standard geometric representation of a Coxeter group. We prove sufficient conditions for faithfulness and non-faithfulness of such a representation. In the case when the weighted Coxeter graph is balanced we discuss how the generalized geometric representation is related to the numbers game played on the Coxeter graph.
The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In: Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists Delta-regular graphs with arbitrarily large geometric thickness. In particular, for all Delta >= 9 and for all large n, there exists a Delta-re…
A set of geometric graphs is {\em geometric-packable} if it can be asymptotically packed into every sequence of drawings of the complete graph $K_n$. For example, the set of geometric triangles is geometric-packable due to the existence of Steiner Triple Systems. When $G$ is the $4$-cycle (or $4$-cycle with a chord), we show that the set of plane drawings of $G$ is geometric-packable. In contrast, the analogous statement is false when $G$ is nearly any other planar Hamiltonian graph (with at most 3 possible exceptions). A convex geometric graph is {\em convex-packable} if it can be asymptotica…
On asymptotic packing of convex geometric and ordered graphs
本文研究凸几何图与有序图的渐近填充问题，证明循环色数至多4的凸几何图及区间色数至多3的有序图均可填充，并说明该界最优。
Counting the number of Hamiltonian cycles that are contained in a geometric graph is {\bf \#P}-complete even if the graph is known to be planar \cite{lot:refer}. A relaxation for problems in plane geometric graphs is to allow the geometric graphs to be 1-plane, that is, each of its edges is crossed at most once. We consider the following question: For any set $P\/$ of $n\/$ points in the plane, how many 1-plane Hamiltonian cycles can be packed into a complete geometric graph $K_n\/$? We investigate the problem by taking two different situations of $P\/$, namely, when $P\/$ is in convex positio…
Geometric structures in pseudo-random graphs
本文提出了一个在伪随机图中计数几何结构的通用框架，恢复并改进了连续问题在有限域类比下的若干结果，揭示了离散几何、几何测度论与图论间的相互作用。
We uncover a connection between two seemingly unrelated notions: lettericity, from structural graph theory, and geometric griddability, from the world of permutation patterns. Both of these notions capture important structural properties of their respective classes of objects. We prove that these notions are equivalent in the sense that a permutation class is geometrically griddable if and only if the corresponding class of inversion graphs has bounded lettericity.
Geometric constructions of small regular graphs with girth 7
该研究利用广义四边形的关联图进行几何构造，得到围长为7的三类正则图无限族，部分图达到目前已知最小阶数。
In this paper we present a complete characterization of the smallest sets which block all the simple perfect matchings in a complete convex geometric graph on $2m$ vertices. In particular, we show that all these sets are caterpillar graphs with a special structure, and that their total number is $m \cdot 2^{m-1}$.
In response to a well-known open question ``Does every complete geometric graph on $2n\/$ vertices have a partition of its edge set into $n\/$ plane spanning trees?" we provide an affirmative answer when the complete geometry graph is in the regular wheel configuration. Also we present sufficient conditions for the complete geometric graph on $2n\/$ vertices to have a partition of its edge set into $n\/$ plane spanning trees (which are double stars, caterpillars or $ w\/$-caterpillars).
Cameron and Kazanidis have recently shown that rank-3 graphs are either cores or have complete cores, and they asked whether this holds for all strongly regular graphs. We prove that this is true for the point graphs and line graphs of generalized quadrangles and that when the number of points is sufficiently large, it is also true for the block graphs of Steiner systems and orthogonal arrays.
We analyse properties of geometric intersection graphs to show the strict containment between some natural classes of geometric intersection graphs. In particular, we show the following properties: - A graph $G$ is outerplanar if and only if the 1-subdivision of $G$ is outer-segment. - For each integer $k\ge 1$, the class of intersection graphs of segments with $k$ different lengths is a strict subclass of the class of intersection graphs of segments with $k+1$ different lengths. - For each integer $k\ge 1$, the class of intersection graphs of disks with $k$ different sizes is a strict subclas…
In this paper, we provide an algorithm for traversing geometric graphs which visits all vertices, and reports every vertex and edge exactly once. To achieve this, we combine a given geometric graph $G$ with the integer lattice, seen as a graph, in such a way that the resulting hypothetical graph can be traversed using the algorithm in \cite{Chavez}. To overcome the problem with hypothetical vertices and edges, we develop an algorithm for visiting any $k$th neighborhood of a vertex in a graph straight-line drawn in the plane using $O(\log k)$ memory. The memory needed to complete the traversal …
Let $G$ be a graph obtained as the union of some $n$-vertex graph $H_n$ with minimum degree $δ(H_n)\geqαn$ and a $d$-dimensional random geometric graph $G^d(n,r)$. We investigate under which conditions for $r$ the graph $G$ will a.a.s. contain the $k$-th power of a Hamilton cycle, for any choice of $H_n$. We provide asymptotically optimal conditions for $r$ for all values of $α$, $d$ and $k$. This has applications in the containment of other spanning structures, such as $F$-factors.
Let P be a finite set of points in general position in the plane. The structure of the complete graph K(P) as a geometric graph includes, for any pair [a,b],[c,d] of vertex-disjoint edges, the information whether they cross or not. The simple (i.e., non-crossing) spanning trees (SSTs) of K(P) are the vertices of the so-called Geometric Tree Graph of P, G(P). Two such vertices are adjacent in G(P) if they differ in exactly two edges, i.e., if one can be obtained from the other by deleting an edge and adding another edge. In this paper we show how to reconstruct from G(P) (regarded as an abstrac…
We use production matrices to count several classes of geometric graphs. We present novel production matrices for non-crossing partitions, connected geometric graphs, and k-angulations, which provide another way of counting the number of such objects. Counting geometric graphs is then equivalent to calculating the powers of a production matrix. Applying the technique of Riordan Arrays to these production matrices, we establish new formulas for the numbers of geometric graphs as well as combinatorial identities derived from the production matrices. Further, we obtain the characteristic polynomi…
A graph is 2-degenerate if every subgraph contains a vertex of degree at most 2. We show that every 2-degenerate graph can be drawn with straight lines such that the drawing decomposes into 4 plane forests. Therefore, the geometric arboricity, and hence the geometric thickness, of 2-degenerate graphs is at most 4. On the other hand, we show that there are 2-degenerate graphs that do not admit any straight-line drawing with a decomposition of the edge set into 2 plane graphs. That is, there are 2-degenerate graphs with geometric thickness, and hence geometric arboricity, at least 3. This answer…
Geometric Graphs with Unbounded Flip-Width
本文证明多种几何图（如区间图、置换图、圆图等）具有无界翻转宽度，该概念由Toruńczyk提出，扩展了图宽度理论的应用范围。
A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called pointed if it lies outside of the convex hull of its neighbours. We show that for a geometric graph with $n$ vertices and $e$ edges there are at least $\frac{n}{2}\binom{2e/n}{3}$ pairs of disjoint edges provided that $2e\geq n$ and all the vertices of the graph are pointed. Besides, we prove that if any edge of a geometric graph with $n$ vertices is disjoint from at most $ m $ edges, then …
We show that a sequence of graphs with geometric property (T) has many small cycles. We also show that when a small part of a sequence of graphs with geometric property (T) is changed, it still has geometric property (T), provided that it is still an expander. We use this to give an example of a sequence of graphs with geometric property (T) that has large cycle-free balls.
The divisor theory for graphs is compared to the theory of linear series on curves through the correspondence associating a curve to its dual graph. An algebro-geometric interpretation of the combinatorial rank is proposed, and proved in some cases.
We define the geometric thickness of a graph to be the smallest number of layers such that we can draw the graph in the plane with straight-line edges and assign each edge to a layer so that no two edges on the same layer cross. The geometric thickness lies between two previously studied quantities, the (graph-theoretical) thickness and the book thickness. We investigate the geometric thickness of the family of complete graphs, K_n. We show that the geometric thickness of K_n lies between ceiling((n/5.646) + 0.342) and ceiling(n/4), and we give exact values of the geometric thickness of K_n fo…
In our previous paper, we presented the combinatorial theory for minimal isostatic pinned frameworks - Assur graphs - which arise in the analysis of mechanical linkages. In this paper we further explore the geometric properties of Assur graphs, with a focus on singular realizations which have static self-stresses. We provide a new geometric characterization of Assur graphs, based on special singular realizations. These singular positions are then related to dead-end positions in which an associated mechanism with an inserted driver will stop or jam.
A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. A geometric graph contains a (k,l)-crossing family if there is a pair of edge subsets E_1,E_2 such that |E_1| = k and |E_2| = l, the edges in E_1 are pairwise crossing, the edges in E_2 are pairwise crossing, and every edges in E_1 is disjoint to every edge in E_2. We conjecture that for any fixed k,l, every n-vertex geometric graph with no (k,l)-crossing family has at most c_{k,l}n edges, where c_{k,l} is a constant that depends only on k and l. In this note, we show that e…
We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is \emph{universal} for a class $\mathcal H$ of planar graphs if it contains an embedding, i.e., a crossing-free drawing, of every graph in $\mathcal H$. Our main result is that there exists a geometric graph with $n$ vertices and $O(n \log n)$ edges that is universal for $n$-vertex forests; this extends to the geometric setting a well-known graph-theoretic result by Chung and Graham, which states…
Given any two vertices u, v of a random geometric graph, denote by d_E(u,v) their Euclidean distance and by d_G(u,v) their graph distance. The problem of finding upper bounds on d_G(u,v) in terms of d_E(u,v) has received a lot of attention in the literature. In this paper, we improve these upper bounds for values of r=omega(sqrt(log n)) (i.e. for r above the connectivity threshold). Our result also improves the best-known estimates on the diameter of random geometric graphs. We also provide a lower bound on d_G(u,v) in terms of d_E(u,v).
On Grundy indices for complete geometric graphs
本文扩展伪Grundy指标至几何图，研究完全几何图的该指标，并给出相关结果。
In this paper, we study orthogonal colourings of random geometric graphs. Two colourings of a graph are orthogonal if they have the property that when two vertices receive the same colour in one colouring, then those vertices receive distinct colours in the other colouring. A random geometric graph $RG(n,r)$ is a graph constructed by randomly placing $n$ vertices in the unit square and connecting two vertices with an edge if and only if their distance is less than the threshold $r$. We show first that random geometric graphs with $r>n^{-α}$, where $0\leq α\leq\frac{1}{4}$, have an orthogonal c…
Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric intersection graphs GB where each body of the collection B is represented by a vertex, and two vertices of GB are adjacent if the intersection of the corresponding bodies is non-empty. For such graph classes and under natural restrictions on their maximum degree or subgraph exclusion, we prove that the relation between their treewidth and the maximum size of a…
Let $G_n$ be a random geometric graph, and then for $q,p \in [0,1)$ we construct a "$(q,p)$-perturbed noisy random geometric graph" $G_n^{q,p}$ where each existing edge in $G_n$ is removed with probability $q$, while and each non-existent edge in $G_n$ is inserted with probability $p$. We give asymptotically tight bounds on the clique number $ω\left(G_n^{q,p}\right)$ for several regimes of parameter.
Thresholds for geometric graphs
本文研究了几何图中阈值的存在性，证明了标准的环面、球面和立方体具有阈值性质。
We consider isomorphism properties of infinite random geometric graphs defined over a variety of metrics. In previous work, it was shown that for $\mathbb{R}^n$ with the $L_{\infty}$-metric, the infinite random geometric graph is, with probability 1, unique up to isomorphism. However, in the case $n=2$ this is false with either of the $L_2$-metric. We generalize this result to a large family of metrics induced by norms. Within this class of metric spaces, we show that the infinite geometric graph is not unique up to isomorphism if it has the new property which we name truncating: each step-iso…
Extremal Chemical Graphs for the Arithmetic-Geometric Index
该研究针对数学化学中新提出的算术-几何指数这一基于度的图不变量，推导了给定阶数与大小的连通化学图的该指数严格上界、刻画了极值图，并证实去除连通性约束无法提升该上界。
We give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for satisfiability.
We introduce a new class of countably infinite random geometric graphs, whose vertices are points in a metric space, and vertices are adjacent independently with probability p if the metric distance between the vertices is below a given threshold. If the vertex set is a countable dense set in R^n equipped with the metric derived from the L_{\infty}-norm, then it is shown that with probability 1 such infinite random geometric graphs have a unique isomorphism type. The isomorphism type, which we call GR_n, is characterized by a geometric analogue of the existentially closed adjacency property, a…
Spanning trees of bounded degree in random geometric graphs
本文确定了随机几何图中包含所有有界度生成树的尖锐阈值，提供了二项随机图结果的几何对应，证实了相关猜想，并给出了适用于有界亏格或树宽图族的算法证明。
For undirected graphs, the Ricci curvature introduced by Lin-Lu-Yau has been widely studied from various perspectives, especially geometric analysis. In the present paper, we discuss generalization problem of their Ricci curvature for directed graphs. We introduce a new generalization by using the mean transition probability kernel which appears in the formulation of the Chung Laplacian. We conclude several geometric and spectral properties of directed graphs under a lower Ricci curvature bound extending previous results in the undirected case.
Finding the Cores of Higher Graphs Using Geometric and Topological Means: A Survey
本综述探讨了利用离散曲率、有效电阻和持久同调等几何与拓扑方法寻找高阶图核的最新文献，旨在连接图论、离散几何和计算拓扑工具，以启发高阶图简化的新研究。
First-Order Logic and Twin-Width for Some Geometric Graphs
本文证明了某些几何图类中一阶逻辑可计算性与双宽度参数的等价性，包括非退化轴对齐单位线段图和圆弧图，但不适用于1.5D地形的可视图。
The main goal of this paper is to formalize and explore a connection between chromatic properties of graphs with geometric representations and competitive analysis of on-line algorithms, which became apparent after the recent construction of triangle-free geometric intersection graphs with arbitrarily large chromatic number due to Pawlik et al. We show that on-line graph coloring problems give rise to classes of game graphs with a natural geometric interpretation. We use this concept to estimate the chromatic number of graphs with geometric representations by finding, for appropriate simpler g…
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