arXiv subjects

Dieter Mitsche

Publications and source records attributed to Dieter Mitsche.

At least 37 records · Page 2Linked to original sources

arXiv2021

On the power of choice for Boolean functions

In this paper we consider a variant of the well-known Achlioptas process for graphs adapted to monotone Boolean functions. Fix a number of choices $r\in \mathbb N$ and a sequence of increasing functions $(f_n)_{n\ge 1}$ such that, for every $n\ge 1$, $f_n:\{0,1\}^n\mapsto \{0,1\}$. Given $n$ bits which are all initially equal to 0, at each step $r$ 0-bits are sampled uniformly at random and are proposed to an agent. Then, the agent selects one of the proposed bits and turns it from 0 to 1 with the goal to reach the preimage of 1 as quickly as possible. We nearly characterize the conditions under which an acceleration by a factor of $r(1+o(1))$ is possible, and underline the wide applicability of our results by giving examples from the fields of Boolean functions and graph theory.

math.CO↗

arXiv2021

The jump of the clique chromatic number of random graphs

The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Pralat noted that around p \approx n^{-1/2} the clique chromatic number of the random graph G_{n,p} changes by n^{Ω(1)} when we increase the edge-probability p by n^{o(1)}, but left the details of this surprising phenomenon as an open problem. We settle this problem, i.e., resolve the nature of this polynomial `jump' of the clique chromatic number of the random graph G_{n,p} around edge-probability p \approx n^{-1/2}. Our proof uses a mix of approximation and concentration arguments, which enables us to (i) go beyond Janson's inequality used in previous work and (ii) determine the clique chromatic number of G_{n,p} up to logarithmic factors for any edge-probability p.

math.CO↗

arXiv2020

The contact process on random hyperbolic graphs: metastability and critical exponents

We consider the contact process on the model of hyperbolic random graph, in the regime when the degree distribution obeys a power law with exponent $χ\in(1,2)$ (so that the degree distribution has finite mean and infinite second moment). We show that the probability of non-extinction as the rate of infection goes to zero decays as a power law with an exponent that only depends on $χ$ and which is the same as in the configuration model, suggesting some universality of this critical exponent. We also consider finite versions of the hyperbolic graph and prove metastability results, as the size of the graph goes to infinity.

math.PR↗

arXiv2020

Limit theory of combinatorial optimization for random geometric graphs

In the random geometric graph $G(n,r_n)$, $n$ vertices are placed randomly in Euclidean $d$-space and edges are added between any pair of vertices distant at most $r_n$ from each other. We establish strong laws of large numbers (LLNs) for a large class of graph parameters, evaluated for $G(n,r_n)$ in the thermodynamic limit with $nr_n^d =$ const., and also in the dense limit with $n r_n^d \to \infty$, $r_n \to 0$. Examples include domination number, independence number, clique-covering number, eternal domination number and triangle packing number. The general theory is based on certain subadditivity and superadditivity properties, and also yields LLNs for other functionals such as the minimum weight for the travelling salesman, spanning tree, matching, bipartite matching and bipartite travelling salesman problems, for a general class of weight functions with at most polynomial growth of order $d-\varepsilon$, under thermodynamic scaling of the distance parameter.

math.PR↗

arXiv2020

On the Decycling Number of $4$-regular Random Graphs

The decycling number $ϕ(G)$ of a graph $G$ is the smallest number of vertices which can be removed from $G$ so that the resulting graph has no cycles. Bau, Wormald and Zhou conjectured that with probability tending to one the decycling number of the random $4$-regular graph $G_4(n)$ on $n$ vertices is equal to $\lceil (n+1)/3\rceil$. In this paper we show that this conjecture holds asymptotically, i.e. asymptotically almost surely $\lim_{n \to \infty}ϕ(G_4(n))/n = 1/3$.

math.PR↗

arXiv2020

On the largest component of subcritical random hyperbolic graphs

We consider the random hyperbolic graph model introduced by [KPK + 10] and then formalized by [GPP12]. We show that, in the subcritical case $α$ > 1, the size of the largest component is n^{1/(2$α$)+o(1)} , thus strengthening a result of [BFM15] which gave only an upper bound of n^{1/$α$+o(1)}.

math.PR↗

arXiv2020

A note on the independence number, domination number and related parameters of random binary search trees and random recursive trees

We identify the mean growth of the independence number of random binary search trees and random recursive trees and show normal fluctuations around their means. Similarly we also show normal limit laws for the domination number and variations of it for these two cases of random tree models. Our results are an application of a recent general theorem of Holmgren and Janson on fringe trees in these two random tree models.

math.PR↗

arXiv2019

Learning random points from geometric graphs or orderings

Suppose that there is a family of $n$ random points $X_v$ for $v \in V$, independently and uniformly distributed in the square $\left[-\sqrt{n}/2,\sqrt{n}/2\right]^2$ of area $n$. We do not see these points, but learn about them in one of the following two ways. Suppose first that we are given the corresponding random geometric graph $G$, where distinct vertices $u$ and $v$ are adjacent when the Euclidean distance $d_E(X_u,X_v)$ is at most $r$. If the threshold distance $r$ satisfies $n^{3/14} \ll r \ll n^{1/2}$, then the following holds with high probability. Given the graph $G$ (without any geometric information), in polynomial time we can approximately reconstruct the hidden embedding, in the sense that, `up to symmetries', for each vertex $v$ we find a point within distance about $r$ of $X_v$; that is, we find an embedding with `displacement' at most about $r$. Now suppose that, instead of being given the graph $G$, we are given, for each vertex $v$, the ordering of the other vertices by increasing Euclidean distance from $v$. Then, with high probability, in polynomial time we can find an embedding with the much smaller displacement error $O(\sqrt{\log n})$.

math.PR↗

arXiv2019

On the second largest component of random hyperbolic graphs

We show that in the random hyperbolic graph model as formalized by Gugelmann et al. in the most interesting range of $\frac12 < α< 1$ the size of the second largest component is $Θ((\log n)^{1/(1-α)})$, thus answering a question of Bode et al. We also show that for $α=\frac12$ with constant probability the corresponding size is $Θ(\log n)$, whereas for $α=1$ it is $Ω(n^{b})$ for some $b > 0$.

math.PR↗

arXiv2019

k-regular subgraphs near the k-core threshold of a random graph

We prove that $G_{n,p=c/n}$ whp has a $k$-regular subgraph if $c$ is at least $e^{-Θ(k)}$ above the threshold for the appearance of a subgraph with minimum degree at least $k$; i.e. an non-empty $k$-core. In particular, this pins down the threshold for the appearance of a $k$-regular subgraph to a window of size $e^{-Θ(k)}$.

math.CO↗

arXiv2019

Hamilton cycles and perfect matchings in the KPKVB model

In this paper we consider the existence of Hamilton cycles and perfect matchings in a random graph model proposed by Krioukov et al.~in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has been previously shown that this model has various properties associated with complex networks, including a power-law degree distribution, "short distances" and a strictly positive clustering coefficient. The model is specified using three parameters: the number of nodes $n$, which we think of as going to infinity, and $α, ν> 0$, which we think of as constant. Roughly speaking $α$ controls the power law exponent of the degree sequence and $ν$ the average degree. Here we show that for every $α< 1/2$ and $ν=ν(α)$ sufficiently small, the model does not contain a perfect matching with high probability, whereas for every $α< 1/2$ and $ν=ν(α)$ sufficiently large, the model contains a Hamilton cycle with high probability.

math.PR↗

arXiv2018

Clique colourings of geometric graphs

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 $x_1, \ldots,x_n$ 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 $x_i$ and $x_j$ 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 $RG$ 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.

math.CO↗

arXiv2018

Optimal Grid Drawings of Complete Multipartite Graphs and an Integer Variant of the Algebraic Connectivity

How to draw the vertices of a complete multipartite graph $G$ on different points of a bounded $d$-dimensional integer grid, such that the sum of squared distances between vertices of $G$ is (i) minimized or (ii) maximized? For both problems we provide a characterization of the solutions. For the particular case $d=1$, our solution for (i) also settles the minimum-2-sum problem for complete bipartite graphs; the minimum-2-sum problem was defined by Juvan and Mohar in 1992. Weighted centroidal Voronoi tessellations are the solution for (ii). Such drawings are related with Laplacian eigenvalues of graphs. This motivates us to study which properties of the algebraic connectivity of graphs carry over to the restricted setting of drawings of graphs with integer coordinates.

cs.DM↗

arXiv2017

Spectral Gap of Random Hyperbolic Graphs and Related Parameters

Random hyperbolic graphs have been suggested as a promising model of social networks. A few of their fundamental parameters have been studied. However, none of them concerns their spectra. We consider the random hyperbolic graph model as formalized by [GPP12] and essentially determine the spectral gap of their normalized Laplacian. Specifically, we establish that with high probability the second smallest eigenvalue of the normalized Laplacian of the giant component of and $n$-vertex random hyperbolic graph is $Ω(n^{-(2α-1)}/D)$, where $\frac12<α<1$ is a model parameter and $D$ is the network diameter (which is known to be at most polylogarithmic in $n$). We also show a matching (up to a polylogarithmic factor) upper bound of $n^{-(2α-1)}(\log n)^{1+o(1)}$. As a byproduct we conclude that the conductance upper bound on the eigenvalue gap obtained via Cheeger's inequality is essentially tight. We also provide a more detailed picture of the collection of vertices on which the bound on the conductance is attained, in particular showing that for all subsets whose volume is $O(n^{1-\varepsilon})$ the obtained conductance is with high probability $Ω(n^{-(2α-1)\varepsilon+o(1)})$. Finally, we also show consequences of our result for the minimum and maximum bisection of the giant component.

math.PR↗

arXiv2016

The Total Acquisition Number of Random Geometric Graphs

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum cardinality of the set of vertices with positive weight at the end of the process. In this paper, we investigate random geometric graphs $G(n,r)$ with $n$ vertices distributed u.a.r. in $[0,\sqrt{n}]^2$ and two vertices being adjacent if and only if their distance is at most $r$. We show that asymptotically almost surely $a_t(G(n,r)) = Θ( n / (r \lg r)^2)$ for the whole range of $r=r_n \ge 1$ such that $r \lg r \le \sqrt{n}$. By monotonicity, asymptotically almost surely $a_t(G(n,r)) = Θ(n)$ if $r < 1$, and $a_t(G(n,r)) = Θ(1)$ if $r \lg r > \sqrt{n}$.

math.CO↗

arXiv2016

Clique colouring of binomial random graphs

A clique colouring of a graph is a colouring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colours in such a colouring is the clique chromatic number. In this paper, we study the asymptotic behaviour of the clique chromatic number of the random graph G(n,p) for a wide range of edge-probabilities p=p(n). We see that the typical clique chromatic number, as a function of the average degree, forms an intriguing step function.

math.PR↗

arXiv2016

Straightness of rectilinear vs. radio-concentric networks: modeling simulation and comparison

This paper proposes a comparison between rectilinear and radio-concentric networks. Indeed, those networks are often observed in urban areas, in several cities all over the world. One of the interesting properties of such networks is described by the \textit{straightness} measure from graph theory, which assesses how much moving from one node to another along the network links departs from the network-independent straightforward path. We study this property in both rectilinear and radio-concentric networks, first by analyzing mathematically routes from the center to peripheral locations in a theoretical framework with perfect topology, then using simulations for multiple origin-destination paths. We show that in most of the cases, radio-concentric networks have a better straightness than rectilinear ones. How may this property be used in the future for urban networks?

cs.SI↗

arXiv2016

The bondage number of random graphs

A dominating set of a graph is a subset $D$ of its vertices such that every vertex not in $D$ is adjacent to at least one member of $D$. The domination number of a graph $G$ is the number of vertices in a smallest dominating set of $G$. The bondage number of a nonempty graph $G$ is the size of a smallest set of edges whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. In this note, we study the bondage number of binomial random graph $G(n,p)$. We obtain a lower bound that matches the order of the trivial upper bound. As a side product, we give a one-point concentration result for the domination number of $G(n,p)$ under certain restrictions.

math.CO↗