scholarly journals Clique Colourings of Geometric Graphs

10.37236/7159 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Colin McDiarmid ◽  
Dieter Mitsche ◽  
Pawel Prałat
Keyword(s):  
Asymptotic Behaviour ◽  
Chromatic Number ◽  
Euclidean Distance ◽  
High Probability ◽  
Maximal Clique ◽  
Higher Dimensions ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Vertex Set

A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number.  Given $n$ points $\mathbf{x}_1, \ldots,\mathbf{x}_n$ in the plane, and a threshold $r>0$, the corresponding geometric graph has vertex set $\{v_1,\ldots,v_n\}$, and distinct $v_i$ and $v_j$ are adjacent when the Euclidean distance between $\mathbf{x}_i$ and $\mathbf{x}_j$ is at most $r$. We investigate the clique chromatic number of such graphs.We first show that the clique chromatic number is at most 9 for any geometric graph in the plane, and briefly consider geometric graphs in higher dimensions. Then we study the asymptotic behaviour of the clique chromatic number for the random geometric graph $\mathcal{G}$ in the plane, where $n$ random points are independently and uniformly distributed in a suitable square. We see that as $r$ increases from 0, with high probability the clique chromatic number is 1 for very small $r$, then 2 for small $r$, then at least 3 for larger $r$, and finally drops back to 2.

Infection Spread in Random Geometric Graphs

2015 ◽  
Vol 47 (1) ◽  
pp. 164-181 ◽  
Author(s):  
Ghurumuruhan Ganesan
Keyword(s):  
Lower Bound ◽  
High Probability ◽  
Contact Process ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Random Geometric Graphs ◽  
Long Time ◽  
Explicit Lower Bound ◽  
Infection Spread

In this paper we study the speed of infection spread and the survival of the contact process in the random geometric graph G = G(n, rn, f) of n nodes independently distributed in S = [-½, ½]2 according to a certain density f(·). In the first part of the paper we assume that infection spreads from one node to another at unit rate and that infected nodes stay in the same state forever. We provide an explicit lower bound on the speed of infection spread and prove that infection spreads in G with speed at least D1nrn2. In the second part of the paper we consider the contact process ξt on G where infection spreads at rate λ > 0 from one node to another and each node independently recovers at unit rate. We prove that, for every λ > 0, with high probability, the contact process on G survives for an exponentially long time; there exist positive constants c1 and c2 such that, with probability at least 1 - c1 / n4, the contact process starting with all nodes infected survives up to time tn = exp(c2n/logn) for all n.

scholarly journals Cops and Robbers on Geometric Graphs

2012 ◽  
Vol 21 (6) ◽  
pp. 816-834 ◽  
Author(s):  
ANDREW BEVERIDGE ◽  
ANDRZEJ DUDEK ◽  
ALAN FRIEZE ◽  
TOBIAS MÜLLER
Keyword(s):  
Lower Bound ◽  
Absolute Constant ◽  
High Probability ◽  
Probability Space ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graphs ◽  
Cops And Robbers ◽  
Minimum Number ◽  
Vertex Set

Cops and robbers is a turn-based pursuit game played on a graph G. One robber is pursued by a set of cops. In each round, these agents move between vertices along the edges of the graph. The cop number c(G) denotes the minimum number of cops required to catch the robber in finite time. We study the cop number of geometric graphs. For points x1, . . ., xn ∈ ℝ2, and r ∈ ℝ+, the vertex set of the geometric graph G(x1, . . ., xn; r) is the graph on these n points, with xi, xj adjacent when ∥xi − xj∥ ≤ r. We prove that c(G) ≤ 9 for any connected geometric graph G in ℝ2 and we give an example of a connected geometric graph with c(G) = 3. We improve on our upper bound for random geometric graphs that are sufficiently dense. Let (n,r) denote the probability space of geometric graphs with n vertices chosen uniformly and independently from [0,1]2. For G ∈ (n,r), we show that with high probability (w.h.p.), if r ≥ K1 (log n/n)1/4 then c(G) ≤ 2, and if r ≥ K2(log n/n)1/5 then c(G) = 1, where K1, K2 > 0 are absolute constants. Finally, we provide a lower bound near the connectivity regime of (n,r): if r ≤ K3 log n/ then c(G) > 1 w.h.p., where K3 > 0 is an absolute constant.

scholarly journals Plane and planarity thresholds for random geometric graphs

2019 ◽  
Vol 12 (01) ◽  
pp. 2050005
Author(s):  
Ahmad Biniaz ◽  
Evangelos Kranakis ◽  
Anil Maheshwari ◽  
Michiel Smid
Keyword(s):  
Euclidean Distance ◽  
Independent Set ◽  
Geometric Graph ◽  
Threshold Function ◽  
Geometric Graphs ◽  
Connected Subgraph ◽  
Random Geometric Graph ◽  
Distance Threshold ◽  
Random Geometric Graphs ◽  
Straight Line

A random geometric graph, [Formula: see text], is formed by choosing [Formula: see text] points independently and uniformly at random in a unit square; two points are connected by a straight-line edge if they are at Euclidean distance at most [Formula: see text]. For a given constant [Formula: see text], we show that [Formula: see text] is a distance threshold function for [Formula: see text] to have a connected subgraph on [Formula: see text] points. Based on this, we show that [Formula: see text] is a distance threshold for [Formula: see text] to be plane, and [Formula: see text] is a distance threshold to be planar. We also investigate distance thresholds for [Formula: see text] to have a non-crossing edge, a clique of a given size, and an independent set of a given size.

Infection Spread in Random Geometric Graphs

2015 ◽  
Vol 47 (01) ◽  
pp. 164-181 ◽  
Author(s):  
Ghurumuruhan Ganesan
Keyword(s):  
Lower Bound ◽  
High Probability ◽  
Contact Process ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Random Geometric Graphs ◽  
Long Time ◽  
Explicit Lower Bound ◽  
Infection Spread

In this paper we study the speed of infection spread and the survival of the contact process in the random geometric graph G = G(n, r n , f) of n nodes independently distributed in S = [-½, ½]2 according to a certain density f(·). In the first part of the paper we assume that infection spreads from one node to another at unit rate and that infected nodes stay in the same state forever. We provide an explicit lower bound on the speed of infection spread and prove that infection spreads in G with speed at least D 1 nr n 2. In the second part of the paper we consider the contact process ξ t on G where infection spreads at rate λ > 0 from one node to another and each node independently recovers at unit rate. We prove that, for every λ > 0, with high probability, the contact process on G survives for an exponentially long time; there exist positive constants c 1 and c 2 such that, with probability at least 1 - c 1 / n 4, the contact process starting with all nodes infected survives up to time t n = exp(c 2 n/logn) for all n.

scholarly journals Colouring random geometric graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Colin J. H. McDiarmid ◽  
Tobias Müller
Keyword(s):  
Chromatic Number ◽  
Analogous Result ◽  
Clique Number ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Random Geometric Graphs ◽  
Different Types ◽  
International Audience ◽  
Uniform Case

International audience A random geometric graph $G_n$ is obtained as follows. We take $X_1, X_2, \ldots, X_n ∈\mathbb{R}^d$ at random (i.i.d. according to some probability distribution ν on $\mathbb{R}^d$). For $i ≠j$ we join $X_i$ and $X_j$ by an edge if $║X_i - X_j ║< r(n)$. We study the properties of the chromatic number $χ _n$ and clique number $ω _n$ of this graph as n becomes large, where we assume that $r(n) →0$. We allow any choice $ν$ that has a bounded density function and $║. ║$ may be any norm on $ℝ^d$. Depending on the choice of $r(n)$, qualitatively different types of behaviour can be observed. We distinguish three main cases, in terms of the key quantity $n r^d$ (which is a measure of the average degree). If $r(n)$ is such that $\frac{nr^d}{ln n} →0$ as $n →∞$ then $\frac{χ _n}{ ω _n} →1$ almost surely. If n $\frac{r^d }{\ln n} →∞$ then $\frac{χ _n }{ ω _n} →1 / δ$ almost surely, where $δ$ is the (translational) packing density of the unit ball $B := \{ x ∈ℝ^d: ║x║< 1 \}$ (i.e. $δ$ is the proportion of $d$-space that can be filled with disjoint translates of $B$). If $\frac{n r^d }{\ln n} →t ∈(0,∞)$ then $\frac{χ _n }{ ω _n}$ tends almost surely to a constant that can be bounded in terms of $δ$ and $t$. These results extend earlier work of McDiarmid and Penrose. The proofs in fact yield separate expressions for $χ _n$ and $ω _n$. We are also able to prove a conjecture by Penrose. This states that when $\frac{n r^d }{ln n} →0$ then the clique number becomes focussed on two adjacent integers, meaning that there exists a sequence $k(n)$ such that $\mathbb{P}( ω _n ∈\{k(n), k(n)+1\}) →1$ as $n →∞$. The analogous result holds for the chromatic number (and for the maximum degree, as was already shown by Penrose in the uniform case).

scholarly journals On the relation between graph distance and Euclidean distance in random geometric graphs

2016 ◽  
Vol 48 (3) ◽  
pp. 848-864 ◽  
Author(s):  
J. Díaz ◽  
D. Mitsche ◽  
G. Perarnau ◽  
X. Pérez-Giménez
Keyword(s):  
Lower Bound ◽  
Euclidean Distance ◽  
Geometric Graph ◽  
Upper Bounds ◽  
Geometric Graphs ◽  
Graph Distance ◽  
Random Geometric Graph ◽  
Random Geometric Graphs ◽  
Connectivity Threshold

Abstract Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=ω(√logn) (that is, 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 dE(u, v) conditional on dE(u, v).

scholarly journals Estimating perimeter using graph cuts

2017 ◽  
Vol 49 (4) ◽  
pp. 1067-1090 ◽  
Author(s):  
Nicolás García Trillos ◽  
Dejan Slepčev ◽  
James von Brecht
Keyword(s):  
Confidence Intervals ◽  
Error Estimates ◽  
Approximation Error ◽  
Higher Dimensions ◽  
Geometric Graph ◽  
Graph Cut ◽  
Average Degree ◽  
Random Geometric Graph ◽  
Low Dimensions ◽  
Variance Estimates

Abstract We investigate the estimation of the perimeter of a set by a graph cut of a random geometric graph. For Ω ⊆ D = (0, 1)d with d ≥ 2, we are given n random independent and identically distributed points on D whose membership in Ω is known. We consider the sample as a random geometric graph with connection distance ε > 0. We estimate the perimeter of Ω (relative to D) by the, appropriately rescaled, graph cut between the vertices in Ω and the vertices in D ∖ Ω. We obtain bias and variance estimates on the error, which are optimal in scaling with respect to n and ε. We consider two scaling regimes: the dense (when the average degree of the vertices goes to ∞) and the sparse one (when the degree goes to 0). In the dense regime, there is a crossover in the nature of the approximation at dimension d = 5: we show that in low dimensions d = 2, 3, 4 one can obtain confidence intervals for the approximation error, while in higher dimensions one can obtain only error estimates for testing the hypothesis that the perimeter is less than a given number.

On the treewidth of random geometric graphs and percolated grids

2017 ◽  
Vol 49 (1) ◽  
pp. 49-60 ◽  
Author(s):  
Anshui Li ◽  
Tobias Müller
Keyword(s):  
Critical Radius ◽  
Geometric Graph ◽  
Giant Component ◽  
Bond Percolation ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Random Geometric Graphs

Abstract In this paper we study the treewidth of the random geometric graph, obtained by dropping n points onto the square [0,√n]2 and connecting pairs of points by an edge if their distance is at most r=r(n). We prove a conjecture of Mitsche and Perarnau (2014) stating that, with probability going to 1 as n→∞, the treewidth of the random geometric graph is 𝜣(r√n) when lim inf r>rc, where rc is the critical radius for the appearance of the giant component. The proof makes use of a comparison to standard bond percolation and with a little bit of extra work we are also able to show that, with probability tending to 1 as k→∞, the treewidth of the graph we obtain by retaining each edge of the k×k grid with probability p is 𝜣(k) if p>½ and 𝜣(√log k) if p<½.

scholarly journals Cliques in high-dimensional random geometric graphs

2020 ◽  
Vol 5 (1) ◽  
Author(s):  
Konstantin E. Avrachenkov ◽  
Andrei V. Bobu
Keyword(s):  
Data Science ◽  
Real Life ◽  
Geometric Graph ◽  
Dimensional Euclidean Space ◽  
Geometric Graphs ◽  
Data Set ◽  
Random Geometric Graph ◽  
Random Geometric Graphs ◽  
Real Life Situation ◽  
Average Vertex

AbstractRandom geometric graphs have become now a popular object of research. Defined rather simply, these graphs describe real networks much better than classical Erdős–Rényi graphs due to their ability to produce tightly connected communities. The n vertices of a random geometric graph are points in d-dimensional Euclidean space, and two vertices are adjacent if they are close to each other. Many properties of these graphs have been revealed in the case when d is fixed. However, the case of growing dimension d is practically unexplored. This regime corresponds to a real-life situation when one has a data set of n observations with a significant number of features, a quite common case in data science today. In this paper, we study the clique structure of random geometric graphs when $$n\rightarrow \infty$$ n → ∞ , and $$d \rightarrow \infty$$ d → ∞ , and average vertex degree grows significantly slower than n. We show that under these conditions, random geometric graphs do not contain cliques of size 4 a. s. if only $$d \gg \log ^{1 + \epsilon } n$$ d ≫ log 1 + ϵ n . As for the cliques of size 3, we present new bounds on the expected number of triangles in the case $$\log ^2 n \ll d \ll \log ^3 n$$ log 2 n ≪ d ≪ log 3 n that improve previously known results. In addition, we provide new numerical results showing that the underlying geometry can be detected using the number of triangles even for small n.

scholarly journals The Asymptotic Size of the Largest Component in Random Geometric Graphs with Some Applications

2014 ◽  
Vol 46 (02) ◽  
pp. 307-324 ◽  
Author(s):  
Ge Chen ◽  
Changlong Yao ◽  
Tiande Guo
Keyword(s):  
Limit Theorem ◽  
Open Cluster ◽  
Geometric Graph ◽  
Open Clusters ◽  
Asymptotic Result ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Random Geometric Graphs ◽  
Asymptotic Size ◽  
The Central Limit Theorem

In this paper we estimate the expectation of the size of the largest component in a supercritical random geometric graph; the expectation tends to a polynomial on a rate of exponential decay. We sharpen the expectation's asymptotic result using the central limit theorem. Similar results can be obtained for the size of the biggest open cluster, and for the number of open clusters of percolation on a box, and so on.

Sign in / Sign up

Export Citation Format

Share Document