I work in combinatorics and probability. My research interests include
random graphs, percolation, discrete and stochastic geometry, random walks, random matrices, deterministic graphs in various flavours (extremal, coloured, fractional, topological), asymptotic and probabilistic combinatorics and combinatorial game theory.
The following is a list of my journal articles, with links to preprint versions in pdf format.
 T. Müller,
Two point concentration in random geometric graphs,
Combinatorica, Volume 28, Issue 5, Pages 529545;[+]
A random geometric graph \(G_n\) is constructed by taking vertices \(X_1, \dots, X_n \in {\mathbb R}^d\) at random (i.i.d. according to some probability distribution \(\nu\)
with a bounded density function) and including an edge between \(X_i\) and
\(X_j\) if \(Xi − Xj < r\) where \(r = r(n) > 0\).
We prove a conjecture of Penrose stating that when \(r = r(n)\) is chosen such that
\(nr^d = o(\ln n)\) then the probability distribution of the clique number \(\omega(G_n)\) becomes concentrated on two consecutive integers and we show that the same holds for a number of other graph parameters including the chromatic number \(\chi(G_n)\).

T. Müller, R.J. Waters,
Circular choosability is rational,
Journal of Combinatorial Theory series B, Volume 99, Issue 5, Pages 801813;
[+]
The circular choosability or circular list chromatic number of a graph is a listversion of the circular chromatic number, introduced by Mohar and studied by several other groups of authors. One of the nice properties that the circular chromatic number enjoys is that it is a rational number for all finite graphs G, and a fundamental question, posed by Zhu and reiterated by several others, is whether the same holds for the circular choosability. In this paper we show that this is indeed the case.

R.W. van der Hofstad, W. Kager, T. Müller,
A local limit theorem for the critical random graph,
Electronic Communications in Probability, Volume 14, Pages 122131;
[+]
We consider the limit distribution of the orders of the k largest components in the ErdösRényi random graph inside the “critical window” for arbitrary k. We prove a local limit theorem for this joint distribution and derive an exact expression for the joint probability density function.

J. Balogh, B. Bollobás, M. Krivelevich, T. Müller, M. Walters,
Hamilton cycles in random geometric graphs,
Annals of Applied Probability, Volume 21, Issue 3, Pages 10531072.
(A preprint
on the arxiv.);
[+]
We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2connected. This answers a question of Penrose.
We also show that in the knearest neighbour model, there is a constant κ such that almost every κconnected graph has a Hamilton cycle.

M. Basavaraju, L.S. Chandran, N. Cohen, F. Havet, T. Müller,
Acyclic edgecolouring of planar graphs, SIAM journal on discrete mathematics, Volume 25, Issue 2, Pages 463478.
(Preliminary version in Eurocomb 2009);
[+]
A proper edgecolouring with the property that every cycle contains edges of at least three distinct colours is called an acyclic edgecolouring. The acyclic chromatic index of a graph \(G\), denoted \(\chi_a'(G)\), is the minimum k such that G admits an acyclic edgecolouring with k colours. We conjecture that if \(G\) is planar and \(\Delta(G)\) is large enough then \(\chi_a'(G) = \Delta(G)\).
We settle this conjecture for planar graphs with girth at least 5.
We also show that \(\chi_a'(G) ≤ \Delta(G) + 12\) for all planar \(G\), which improves a previous result by Fiedorowicz et al.
 R.A. Hauser, T. Müller,
Conditioning of random conic systems under a general family of input distributions,
Foundations of Computational Mathematics, Volume 9, Issue 3, Pages 335358.
(A preprint version in the Oxford University Numerical Analysis techreport series  with some results that did not make it to the
final version.);
[+]
We consider the conic feasibility problem associated with the linear homogeneous system
\(Ax \leq 0, x\neq 0\). The complexity of iterative algorithms for solving this problem depends on a condition number \({\cal C}(A)\). When studying the typical behaviour of algorithms under stochastic input one is therefore naturally led to investigate the fatness of the tails of the distribution of \({\cal C}(A)\). Introducing the very general class of uniformly absolutely continuous probability models for the random matrix \(A\), we show that the distribution tails of \({\cal C}(A)\) decrease at algebraic rates, both for the GoffinCheungCucker number \({\cal C}_G\) and the Renegar number \({\cal C}_R\). The exponent that drives the decay arises naturally in the theory of uniform absolute continuity, which we also develop in this paper. In the case of CG we also discuss lower bounds on the tail probabilities and show that there exist absolutely continuous input models for which the tail decay is subalgebraic.

L. AddarioBerry, R.J. Kang, T. Müller,
Acyclic dominating partitions,
Journal of Graph Theory, Volume 64, Issue 5, Pages 292311;
[+]
Given a graph \(G = (V,E)\), let \(P\) be a partition of \(V\). We say that \(P\) is dominating if, for each part \(P\) of \(P\), the set \(V \setminus P\) is a dominating set in \(G\) (equivalently, if every vertex has a neighbour of a different colour from its own). We say that \(P\) is acyclic if for any parts \(P,P′\) of \(P\), the bipartite subgraph \(G[P,P′]\) consisting of the edges between \(P\) and \(P′\) in \(P\) contains no cycles. The acyclic dominating number \(\text{ad}(G)\) of \(G\) is the least number of parts in any partition of \(V\) that is both acyclic and dominating; and we shall denote by \(\text{ad}(d)\) the maximum over all graphs \(G\) of maximum degree at most \(d\)
of \(\text{ad}(G)\). In this paper, we prove that \(\text{ad}(3) = 2\), which establishes a conjecture of Boiron, Sopena and Vignal. For general \(d\), we prove the upper bound \(\text{ad}(d) = O(d \ln d)\) and a lower bound of \(\text{ad}(d) = \Omega(d)\).

R.J. Kang, T. Müller,
Frugal, acyclic and star colourings of graphs,
Discrete Applied Mathematics, Volume 159, Issue 16, Pages 18061814;
[+]
Given a graph \(G = (V, E)\), a vertex colouring of \(V\) is \(t\)frugal if no colour appears more than \(t\) times in any neighbourhood and is acyclic if each of the bipartite graphs consisting of the edges between any two colour classes is acyclic. For graphs of bounded maximum degree, Hind, Molloy and Reed studied proper \(t\)frugal colourings and Yuster studied acyclic proper 2frugal colourings. In this paper, we expand and generalise this study.

M. Bradonjic, T. Müller, A. G. Percus,
Coloring geographical threshold graphs,
Discrete Mathematics & Theoretical Computer Science, Vol 12, No 3, Pages 103114;
[+]
We propose a coloring algorithm for sparse random graphs generated by the geographical threshold
graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. The motivation for analyzing this
model is that many real networks (e.g., wireless networks, the Internet, etc.) need to be studied by using
a “richer” stochastic model (which in this case includes both a distance between nodes and weights on the
nodes). Here, we analyze the GTG coloring algorithm together with the graph’s clique number, showing
formally that in spite of the differences in structure between GTG and RGG, the asymptotic behavior
of the chromatic number is identical: \(\chi = \ln n (1 + 𝑜(1))\). Finally, we consider the leading corrections \(\ln\ln n\) to this expression, again using the coloring algorithm and clique number to provide bounds on the chromatic number. We show that the gap between the lower and upper bound is within \(C \ln n / (\ln \ln n)^2\), and specify the constant \(C\).

T. Müller, A. Pór, J.S. Sereni,
Graphs with four boundary vertices, Electronic journal of Combinatorics, Volume 18, Issue 1, Paper 11, 18 pages;
[+]
A vertex \(v\) of a graph G is a boundary vertex if there exists a vertex \(u\) such that the distance in \(G\) from \(u\) to \(v\) is at least the distance from \(u\) to any neighbour of \(v\). We give a full description of all graphs that have exactly four boundary vertices, which answers a question of Hasegawa and Saito. To this end, we introduce the concept of frame of a graph. It allows us to construct, for every positive integer \(b\) and every possible “distancevector” between b points, a graph \(G\) with exactly \(b\) boundary vertices such that every graph with b boundary vertices and the same distancevector between them is an induced subgraph of G.
 F. Havet, R.J. Kang, T. Müller, J.S. Sereni,
Circular choosability, Journal of Graph Theory, Volume 61, Issue 4 , Pages 241  270;
[+]
We study circular choosability, a notion recently introduced by Mohar and by Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that \(\text{cch}(G) = O(\text{ch}(G) + \ln V (G))\) for every graph \(G\).
We investigate a generalisation of circular choosability, the circular \(f\)choosability, where \(f\) is a function of the degrees. We also consider the circular choice number of planar graphs. Mohar asked for the value of \(\tau := \sup\{\text{cch}(G) : G\text{ is planar}\}\), and we prove that \(6 \leq \tau \leq 8\), thereby providing a negative answer to another question of Mohar. We also study the circular choice number of planar and outerplanar graphs with prescribed girth, and graphs with bounded density.
 T. Müller, A. Pór, J.S. Sereni,
Bounding the boundary by the minimum and the maximum degree, Discrete Mathematics, Volume 308, Issue 24, Pages 65816583;
[+]
A vertex \(v\) of a graph \(G\) is a boundary vertex if there exists a vertex \(u\) such that the distance in \(G\) from \(u\) to \(v\) is at least the distance from \(u\) to any neighbour of \(v\).
We give the best possible lower bound, up to a constant factor, on the number of boundary vertices of a graph in terms of its minimum degree (or maximum degree). This settles a problem introduced by Hasegawa and Saito.
 T. Müller, J.S. Sereni,
Identifying and locatingdominating codes in (random) geometric networks, Combinatorics, Probability and Computing, Volume 18, Issue 6, Pages 925952;
[+]
We model a problem about networks built from wireless devices using identifying and
locatingdominating codes in unit disk graphs. It is known that minimising the size of
an identifying code is NPcomplete even for bipartite graphs. First, we improve this
result by showing that the problem remains NPcomplete for bipartite planar unit
disk graphs. Then, we address the question of the existence of an identifying code for
random unit disk graphs. We derive the probability that there exists an identifying
code as a function of the radius of the disks and we find that for all interesting
ranges of r this probability is bounded away from one. The results obtained are
in sharp contrast with those concerning random graphs in the ErdösRényi model.
Another wellstudied class of codes are locatingdominating codes, which are less
demanding than identifying codes. A locatingdominating code always exists, but
minimising its size is still NPcomplete in general. We extend this result to our
setting by showing that this question remains NPcomplete for arbitrary planar unit
disk graphs. Finally, we study the minimum size of such a code in random unit disk
graphs, and we prove that with probability tending to one, it is of size \((\frac{n}{r})^{2/3+o(1)}\)
if \(r\) is chosen such that \(nr^2 \to \infty\) and of size \(n^{1+o(1)}\) if \(nr^2 \ll \ln n\).
 R.J. Kang, T. Müller, J.S. Sereni,
Improper colouring of (random) unit disk graphs,
Discrete Mathematics, Volume 308, Issue 8, Pages 14381454;
[+]
For any graph G, the \(k\)improper chromatic number \(\chi^k(G)\) is the smallest
number of colours used in a colouring of \(G\) such that each colour class induces a subgraph of maximum degree \(k\). We investigate \(\chi^k\) for unit disk graphs and random unit disk graphs to generalise results of McDiarmid, McDiarmid and Reed and McDiarmid and Müller.

T. Müller, X. PérezGiménez, N. Wormald,
Disjoint Hamilton cycles in the random geometric graph,
Journal of Graph Theory, Volume 68, Issue 4, pages 299–322;
[+]
We consider the standard random geometric graph process in which \(n\) vertices are placed at random on the unit square and edges are sequentially added in increasing order of edgelength. For fixed \(k \geq 1\), we prove that the first edge in the process that creates a \(k\)connected graph coincides a.a.s. with the first edge that causes the graph to contain \(k/2\) pairwise edgedisjoint Hamilton cycles (for even \(k\)), or \((k − 1)/2\) Hamilton cycles plus one perfect matching, all of them pairwise edgedisjoint (for odd \(k\)). This proves and extends a conjecture of Krivelevich and Müller. In the special when case \(k = 2\), our result says that the first edge that makes the random geometric graph Hamiltonian is a.a.s. exactly the same one that gives 2connectivity, which answers a question of Penrose. We prove our results with lengths measured using the \(\ell_p\)norm for any \(p > 1\), and we also extend our result to higher dimensions.

R.J. Kang, L. Lovász, T. Müller, E.R. Scheinerman,
Dot product representations of planar graphs, Electronic Journal of Combinatorics, Volume 18, Issue 1, Paper 216, 14 pages.
(preliminary version in GD 2010);
[+]
A graph \(G\) is a \(k\)dot product graph if there exists a vector labelling \(u : V (G) \to {\mathbb R}^k\) such that \(u(i)^Tu(j) \geq 1\) if and only if \(ij \in E(G)\). Fiduccia, Scheinerman, Trenk and Zito asked whether every planar graph is a 3dot product graph. We show that the answer is “no”. On the other hand, every planar graph is a 4dot product graph. We also answer the corresponding questions for planar graphs of prescribed girth and for outerplanar graphs.
 C.J.H. McDiarmid, T. Müller,
On the chromatic number of random geometric graphs,
Combinatorica, Volume 31, Number 4, Pages 423488;
[+]
Given independent random points \(X_1, \dots, X_n \in {\mathbb R}^d\) with common probability
distribution \(\nu\), and a positive distance \(r = r(n) > 0\), we construct a random geometric
graph \(G_n\) with vertex set \(\{1, . . . , n\}\) where distinct \(i\) and \(j\) are adjacent when \(X_i −X_j \leq r\). Here \(.\) may be any norm on \({\mathbb R}^d\), and \(\nu\) may be any probability distribution
on \({\mathbb R}^d\) with a bounded density function.
We consider the chromatic number \(\chi(G_n)\)
of \(G_n\) and its relation to the clique number \(\omega(G_n)\) as \(n \to \infty\).
Both McDiarmid and Penrose considered the range of \(r\) when
\(r \ll (\frac{\ln n}{n})^{1/d}\) and the range when \(n\)
\(r \gg (\frac{\ln n}{n})^{1/d}\), and their results showed a dramatic difference between these two cases. Here we sharpen and extend the earlier results, and in particular we consider the ‘phase change’ range when \(r \sim (\frac{t \ln n}{n} )^{1/d}\) with \(t > 0\) a fixed constant. Both McDiarmid
and Penrose asked for the behaviour of the chromatic number in this range.
We determine constants \(c(t)\) such that \(\frac{\chi(G_n)}{n r^d} \to c(t)\) almost surely.
Further, we find a “sharp threshold” (except for less interesting choices of the norm when the unit ball tiles \(d\)space): there is a constant \(t_0 > 0\) such that if \(t \leq t_0\) then \(\frac{\chi(G_n)}{\omega(G_n)}\) tends to 1 almost
surely, but if \(t > t_0\) then \(\chi(G_n)\) tends to a limit > 1 almost surely.

R.J. Kang, T. Müller,
Sphere and dot product
representations of graphs, Discrete and Computational Geometry, Volume 47, Number 3, Pages 548568.
(Preliminary version in SoCG 2011);
[+]
A graph \(G\) is a \(k\)sphere graph if there are \(k\)dimensional real vectors \(v_1,\dots,v_n\) such that \(ij \in E(G)\) if and only if the distance between \(v_i\) and \(v_j\) is at most 1.
A graph G is a \(k\)dot product graph if there are \(k\)dimensional real vectors \(v_1, \dots, v_n\) such that \(ij \in E(G)\) if and only if the dot product of \(v_i\) and \(v_j\) is at least 1.
By relating these two geometric graph constructions to oriented \(k\)hyperplane arrangements, we prove that the problems of deciding, given a graph \(G\), whether \(G\) is a \(k\)sphere or a \(k\)dot product graph are NPhard for all \(k > 1\). In the former case, this proves a conjecture of Breu and Kirkpatrick (Comput. Geom. 9:3–24, 1998). In the latter, this answers a question of Fiduccia et al. (Discrete Math. 181:113–138, 1998).
Furthermore, motivated by the question of whether these two recognition problems are in NP, as well as by the implicit graph conjecture, we demonstrate that, for all \(k > 1\), there exist \(k\)sphere graphs and \(k\)dot product graphs such that each representation in \(k\)dimensional real vectors needs at least an exponential number of bits to be stored in the memory of a computer. On the other hand, we show that exponentially many bits are always enough. This resolves a question of Spinrad (Efficient Graph Representations, 2003)
 A. Beveridge, A. Dudek, A. Frieze, T. Müller,
Cops and robbers on geometric graphs,
Combinatorics, Probability and Computing, Volume 21, Issue 06, Pages 816834;
[+]
Cops and robbers is a turnbased 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 \(x_1, \dots, x_n \in {\mathbb R}^2\), and \(r \in {\mathbb R}_+\), the vertex set of the geometric graph \(G(x_1,\dots, x_n; r)\) is the graph on these \(n\) points,
with \(x_i, x_j\) adjacent when \(x_i − x_j \leq r\). We prove that \(c(G) ≤ 9\) for any connected geometric graph \(G\) in \({\mathbb R}^2\) 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 \(G(n,r)\) denote the probability space of geometric graphs with \(n\) vertices chosen uniformly and independently from \([0,1]^2\). For \(G \in G(n,r)\), we show that with high probability (whp), if \(r \geq K_1(\log n/n)^{1/4}\), then \(c(G) \leq 2\), and if \(r \geq K_2(\log n/n)^{1/5}\),
then \(c(G) = 1\) where \(K_1,K_2 > 0\) are absolute constants. Finally, we provide a lower bound near the connectivity regime of \(G(n, r)\): if \(r ≤ K_3 \log n/ \sqrt{n}\) then \(c(G) > 1\) whp, where \(K_3 > 0\) is an absolute constant.

R.J. Kang, M. Mnich, T. Müller,
Induced matchings in subcubic planar graphs, SIAM Journal on Discrete Mathematics, Volume 26, Issue 3, Pages 1383–1411.
(preliminary version in ESA 2010);
[+]
We present a lineartime algorithm that, given a planar graph with \(m\) edges and maximum degree 3, finds an induced matching of size at least \(m/9\). This is best possible.

C.J.H. McDiarmid, T. Müller,
Integer realizations of disk
and segment graphs, Journal of Combinatorial Theory Series B,
Volume 103, Issue 1, Pages 114–143.
(Preliminary version in WG 2010);
[+]
A disk graph is the intersection graph of disks in the plane, a unit disk graph is the intersection graph of same radius disks in the plane, and a segment graph is an intersection graph of line segments in the plane. Every disk graph can be realized by disks with centers on the integer grid and with integer radii; and similarly every unit disk graph can be realized by disks with centers on the integer grid and equal (integer) radius; and every segment graph can be realized by segments whose endpoints lie on the integer grid. Here we show that there exist disk graphs on \(n\) vertices such that in every realization by integer disks at least one coordinate or radius is \(2^{2^{\Omega(n)}}\) and on the other hand every disk graph can be realized by disks with integer coordinates and radii that are at most \(2^{2^{O(n)}}\) ; and we show the analogous results for unit disk graphs and segment graphs. For (unit) disk graphs this answers a question of Spinrad, and for segment graphs this improves over a previous result by Kratochvil and Matousek.

T. Müller, E. J. van Leeuwen, J. van Leeuwen,
Integer Representations of Convex
Polygon Intersection Graphs,
SIAM Jounal on Discrete Mathematics, Volume 27, Issue 1, Pages 205–231.
(Preliminary version in SoCG 2011);
[+]
We determine tight bounds on the smallest size integer grid needed to represent the \(n\)node intersection graphs of a convex polygon \(P\), with \(P\) given in rational coordinates. The intersection graphs only use polygons that are geometrically similar to \(P\) (translates or homothets) and must be represented such that each corner of each polygon lies on a point of the grid. We show the following generic results:
 if \(P\) is a parallelogram and only translates of \(P\) are used, then a \(\Omega(n^2) \times \Omega(n^2)\) grid is sufficient, and needed for some graphs.
 if \(P\) is any other convex polygon and only translates of \(P\) are used, then a
\(2^{\Omega(n)}\times 2^{\Omega(n)}\) grid is sufficient, and needed for some graphs.
 if \(P\) is any convex polygon and arbitrary homothets of \(P\) are allowed, then a \(2^{\Omega(n)}\times 2^{Omega(n)}\) grid is sufficient, and needed for some graphs.
The results substantially improve earlier bounds, and settle the complexity of representing convex polygon intersection graphs. The results also imply small polynomial certificates for the recognition problem for all graph classes considered.

C.J.H. McDiarmid, T. Müller,
The number of disk graphs, European Journal of Combinatorics, Volume 35, Pages 413–431;
[+]
A disk graph is the intersection graph of disks in the plane, and a unit disk graph is the intersection graph of unit radius disks in the plane. We give upper and lower bounds on the number of labelled unit disk and disk graphs on \(n\) vertices. We show that the number of unit disk graphs on \(n\) vertices is \(n^{2n} \cdot \alpha(n)^n\) and the number of disk graphs on n vertices is \(n^{3n} \cdot \beta(n)^n\), where \(\alpha(n)\) and \(\beta(n)\) are \(\Theta(1)\). We conjecture that there exist constants \(\alpha,\beta\) such that the number of unit disk graphs is \(n^{2n} \cdot (\alpha + o(1))^n\) and the number of disk graphs is \(n^{3n} \cdot (\beta + o(1))^n\).
 T. Müller,
A counterexample to a conjecture of Grünbaum on piercing convex sets in the plane,
Discrete Mathematics, Volume 313, Issue 24, Pages 2868–2871;
[+]
A collection of sets \(\cal F\) has the \((p,q)\)property if out of every \(p\) elements of \(\cal F\) there are \(q\) that have a point in common. A transversal of a collection of sets \(\cal F\) is a set \(A\) that intersects every member of \(\cal F\). Grünbaum conjectured that every family \(\cal F\) of closed, convex sets in the plane with the \((4, 3)\)property and at least two elements that are compact has a transversal of bounded cardinality. Here we construct a counterexample to his conjecture. On the positive side, we also show that if such a collection \(\cal F\) contains two disjoint compacta then there is a transveral of cardinality at most 13.

R.J. Kang, T. Müller,
Arrangements of pseudocircles and circles, Discrete and Computational Geometry, Volume 51, Issue 4, Pages 896925;
[+]
An arrangement of pseudocircles is a finite collection of Jordan curves in the plane with the additional properties that (i) every two curves meet in at most two points; and (ii) if two curves meet in a point \(p\), then they cross at \(p\).
We say that two arrangements \({\cal C} = (c_1,\dots,c_n)\) and \({\cal D} = (d_1,\dots,d_n)\) are equivalent if there is a homeomorphism \(\varphi\) of the plane onto itself such that \(\varphi[c_i] = d_i\) for all \(i \in \{1, \dots , n\}\). Linhart and Ortner (2005) gave an example of an arrangement of five pseudocircles that is not equivalent to an arrangement of circles, and they conjectured that every arrangement of at most four pseudocircles is equivalent to an arrangement of circles. Here we prove their conjecture.
We consider two related recognition problems. The first is the problem of deciding, given a (combinatorial description of a) pseudocircle arrangement, whether it is equivalent to an arrangement of circles. The second is deciding whether it is equivalent to an arrangement of convex pseudocircles. We prove that both problems are NPhard, answering questions of Bultena, Grünbaum and Ruskey (1998) and of Linhart and Ortner (2008).
We also give an example of an arrangement of convex pseudocircles with the property that its intersection graph (i.e. the graph with one vertex for each pseudocircle and an edge between two vertices if and only if the corresponding pseudocircles intersect) cannot be realised as the intersection graph of a family of circles. This disproves a folklore conjecture communicated to us by Pyatkin.
 T. Müller, M. Stojakovic,
A threshold for the MakerBreaker Clique game,
Random Structures and Algorithms, Volume 45, Issue 2, Pages 318341;
[+]
We study the MakerBreaker \(k\)clique game played on the edge set of the random graph \(G(n,p)\). In this game, two players, Maker and Breaker, alternately claim unclaimed edges of \(G(n, p)\), until all the edges are claimed. Maker wins if he claims all the edges of a \(k\)clique; Breaker wins otherwise. We determine that the threshold
for the graph property that Maker can win this game is at \(n^{−2/(k+1)}\), for all
\(k > 3\), thus proving a conjecture of Stojakovic and Szabo.
More precisely, we conclude that there exist
constants \(c, C > 0\) such that when \(p > C n^{−2/(k+1)}\) the game is Maker’s win a.a.s., and when \(p < cn^{−2/(k+1)}\) it is Breaker’s win a.a.s.
For the triangle game, when \(k = 3\), we give a more precise result, describing the hitting time of Maker’s win in the random graph process. We show that, with high probability, Maker can win the triangle game exactly at the time when a copy of \(K_5\) with one edge removed appears in the random graph process. As a consequence, we are able to give an expression for the limiting probability of Maker’s win in the triangle game played on the edge set of \(G(n, p)\).

A. Beveridge, A. Dudek, A. Frieze, T. Müller, M. Stojakovic,
MakerBreaker games on random geometric graphs, Random Structures and Algorithms, Volume 45, Issue 4, Pages 553–607;
[+]
In a MakerBreaker game on a graph \(G\), Breaker and Maker alternately claim edges of \(G\). Maker wins if, after all edges have been claimed, the graph induced by his edges has some desired property. We consider four MakerBreaker games played on random geometric graphs. For each of our four games we show that if we add edges between \(n\) points chosen uniformly at random in the unit square by order of increasing edgelength then, with probability tending to one as \(n\to\infty\), the graph becomes Makerwin the very moment it satisfies a simple necessary condition. In particular, with high probability, Maker wins the connectivity game as soon as the minimum degree is at least two; Maker wins the Hamilton cycle game as soon as the minimum degree is at least four; Maker wins the perfect matching game as soon as the minimum degree is at least two and every edge has at least three neighbouring vertices; and Maker wins the Hgame as soon as there is a subgraph from a finite list of “minimal graphs”. These results also allow us to give precise expressions for the limiting probability that \(G(n, r)\) is Makerwin in each case, where \(G(n, r)\) is the graph on n points chosen uniformly at random on the unit square with an edge between two points if and only if their distance is at most \(r\).

R.J. Kang, T. Müller, D.B. West,
On rdynamic coloring of grids, Discrete Applied Mathematics, Volume 186, Pages 286–290;
[+]
An \(r\)dynamic \(k\)coloring of a graph G is a proper \(k\)coloring of \(G\) such that every vertex in \(V(G)\) has neighbors in at least \(\min\{d(v),r\}\) different color classes. The \(r\)dynamic chromatic number of a graph \(G\), written \(\chi_r(G)\), is the least \(k\) such that \(G\) has such a coloring. Proving a conjecture of Jahanbekam, Kim, O, and West, we show that the \(m\)by\(n\) grid has no 3dynamic 4coloring when
\(mn \equiv 2 \text{mod} 4\). This completes the determination of the \(r\)dynamic chromatic number of the \(m\)by\(n\) grid for all \(r,m,n\).

T. Müller, P. Pralat,
The acquaintance time of (percolated) random geometric graphs, European Journal of Combinatorics, Volume 48, Pages 198–214;
[+]
In this paper, we study the acquaintance time \(\text{AC}(G)\) defined for a connected graph \(G\). We focus on \(G(n, r, p)\), a random subgraph of a random geometric graph in which \(n\) vertices are chosen uniformly at random and independently from \([0, 1]^2\), and two vertices are adjacent with probability \(p\) if the Euclidean distance between them is at most \(r\). We present asymptotic results for the acquaintance time of \(G(n,r,p)\) for a wide range of \(p = p(n)\) and \(r = r(n)\). In particular, we show that with high probability \(\text{AC}(G) = \Theta(r^{2})\) for \(G \in G(n, r, 1)\), the “ordinary” random geometric graph, provided that \(\pi nr^2 − \ln n \to \infty\) (that is, above the connectivity threshold). For the percolated random geometric graph \(G \in G(n, r, p)\), we show that with high probability \(\text{AC}(G) = \Theta(r^{−2}p^{−1}\ln n)\), provided that \(pnr^2 \geq n^{1/2+\varepsilon}\) and \(p<1−\varepsilon\) for some
\(\varepsilon >0\).

M. Bode, N. Fountoulakis, T. Müller,
On the largest component of a hyperbolic model of complex networks, Electronic Journal of Combinatorics, Volume 22, Issue 3, Paper P3.24, 43 pages;
[+]
We consider a model for complex networks that was introduced by Krioukov et al. In this model, \(N\) points are chosen randomly inside a disk on the hyperbolic plane and any two of them are joined by an edge if they are within a certain hyperbolic distance. The N points are distributed according to a quasiuniform distribution, which is a distorted version of the uniform distribution. The model turns out to behave similarly to the well known ChungLu model, but without the independence between the edges. Namely, it exhibits a powerlaw degree sequence and small distances but, unlike the ChungLu model and many other wellknown models for complex networks, it also exhibits clustering.
The model is controlled by two parameters \(\alpha\) and \(\nu\) where, roughly speaking, α controls the exponent of the powerlaw and ν controls the average degree. The present paper focuses on the evolution of the component structure of the random graph. We show that (a) for \(\alpha>1\) and \(\nu\) arbitrary, with high probability, as the number of vertices grows, the largest component of the random graph has sublinear order; (b) for \(\alpha < 1\) and \(\nu\) arbitrary with high probability there is a “giant” component of linear order, and (c) when \(\alpha=1\) then there is a nontrivial phase transition for the existence of a linearsized component in terms of \(\nu\).

A. Li, T. Müller, P. Pralat,
Chasing robbers on percolated random geometric graphs, Contributions to Discrete Mathematics, Volume 10, Issue 1, 10 Pages;
[+]
In this paper, we study the vertex pursuit game of Cops and Robbers, in which cops try to capture a robber on the vertices of a graph. The minimum number of cops required to win on a given graph G is called the cop number of \(G\). We focus on \(G(n,r,p)\), a percolated random geometric graph in which \(n\) vertices are chosen uniformly at random and independently from \([0,1]^2\). Two vertices are adjacent with probability \(p\) if the Euclidean distance between them is at most \(r\). If the distance is bigger then \(r\) then they are never adjacent. We present asymptotic results for the game of Cops and Robber played on \(G(n,r,p)\) for a wide range of \(p = p(n)\) and \(r = r(n)\).

T. Müller, R. Spöhel,
A geometric Achlioptas process, Annals of Applied Probability, Volume 25, Issue 6, Pages 32953337;
[+]
The random geometric graph is obtained by sampling \(n\) points from the unit square (uniformly at random and independently), and connecting two points whenever their distance is at most \(r\), for some given \(r = r(n)\). We consider the following variation on the random geometric graph: in each of \(n\) rounds in total, a player is offered two random points from the unit square, and has to select exactly one of these two points for inclusion in the evolving geometric graph.
We study the problem of avoiding a linearsized (or “giant”) component in this setting. Specifically, we show that for any \(r \ll (n \log \log n)^{−1/3}\) there is a strategy that succeeds with high probability in keeping all component sizes sublinear. We also show that this is tight in the following sense: for any \(r \gg (n \log \log n)^{−1/3}\) , with high probabiliy the player will be forced to create a component of size \((1 − o(1))n\), no matter how he plays. We also prove that the corresponding offline problem exhibits a similar threshold behavior at \(r(n) = \Theta(n^{−1/3})\).
These findings should be compared to the existing results for the (ordinary) random geometric graph: there a giant component arises with high probability once \(r\) is of order \(n^{−1/2}\). Thus our results show, in particular, that in the geometric setting the power of choices can be exploited to a much larger extent than in the classical ErdösRényi random graph, where the appearance of a giant component can only be delayed by a constant factor.

M. Bode, N. Fountoulakis, T. Müller,
The probability of connectivity in a hyperbolic model of complex networks, Random Structures and Algorithms, Volume 49, Issue 1, pages 65–94;
[+]
We consider a model for complex networks that was introduced by Krioukov et al. In this model, \(N\) points are chosen randomly inside a disk on the hyperbolic plane according to a distorted version of the uniform distribution and any two of them are joined by an edge if they are within a certain hyperbolic distance. This model exhibits a powerlaw degree sequence, small distances and clustering. The model is controlled by two parameters \(\alpha\) and \(\nu\) where, roughly speaking, \(\alpha\) controls the exponent of the powerlaw and \(\nu\) controls the average degree.
In this paper we focus on the probability that the graph is connected. We show the following results. For \(\alpha>1/2\) and \(\nu\) arbitrary, the graph is disconnected with high probability. For \(\alpha < 1/2\) and \(\nu\) arbitrary, the graph is connected with high probability.
When \(\alpha=1/2\) and \(\nu\) is fixed then the probability of being connected tends to a constant \(f(\nu)\) that depends only on \(\nu\), in a continuous manner. Curiously, \(f(\nu) = 1\) for \(\nu \geq \pi\) while it is strictly increasing, and in particular bounded away from zero and one, for \(0 < \nu < \pi\).

A. Li, T. Müller,
On the treewidth of random geometric graphs and percolated grids, Advances in Applied Probability, Volume 49, Issue 1, pages 4960;
[+]
In this paper, we consider the treewidth of two related random graph models.
The first one is the random geometric graph, where we drop \(n\) points onto
the square \([0,\sqrt{n}]^2\) and connect pairs of points by an edge if their distance is at most \(r = r(n)\).
The second one is the percolated grid, in which we take a \(k\times k\)grid and keep each edge with probability \(p\), independently of all other edges.
We show that, with probability tending to one as \(k \to \infty\), the treewidth of the percolated grid is \(\Theta( k )\) if
\(p>1/2\) and \(\Theta( \sqrt{\log k} )\) if \(p < 1/2\).
By reusing part of the proof of this last result, we also prove a conjecture of Mitsche and Perarnau stating that,
with probability going to one as \(n\to\infty\), the treewidth of the random geometric graph is \(\Theta( r\sqrt{n} )\) when \(\liminf r>r_c\),
where \(r_c\) is the threshold radius for the appearance of the giant component.

T. Müller,
The critical probability for confetti percolation equals 1/2, Random Structures and Algorithms, Volume 50, Issue 4, pages 679697;
[+]
In the confetti percolation model, or twocoloured dead leaves model, radius one disks arrive on the plane according to a spacetime Poisson process. Each disk is coloured back with probability \(p\) and white with probability \(1−p\). In this paper we show that the critical probability for confetti percolation equals 1/2. That is, if \(p > 1/2\) then a.s. there is an unbounded curve in the plane all of whose points are black; while if \(p \leq 1/2\) then a.s. all connected components of the set of black points are bounded. This answers a question of Benjamini and Schramm.

W. Cames van Batenburg, L. Esperet, T. Müller,
Coloring Jordan regions and curves, SIAM Journal on Discrete Mathematics, Volume 31, Issue 3, pages 16701684;
[+]
A Jordan region is a subset of the plane that is homeomorphic to a
closed disk. Consider a family \(\mathcal F\) of Jordan regions whose interiors are
pairwise disjoint, and such that any two Jordan regions intersect in at
most one point. If any point of the plane is contained in at most
\(k\) elements of \(\mathcal F\) (with \(k\) sufficiently large), then we show that the elements of \(\mathcal F\) can be
colored with at most \(k+1\) colors so that intersecting Jordan regions
are assigned distinct colors. This is best possible and answers a question raised by Reed and Shepherd in 1996. As a simple corollary, we also obtain a positive answer to a
problem of Hlineny (1998) on the chromatic number of contact systems of strings.
We also investigate the chromatic number of families of touching Jordan curves. This can be used to bound the ratio between the maximum number of vertexdisjoint directed cycles in a planar digraph, and its fractional counterpart.

N. Fountoulakis, T. Müller,
Law of large numbers for the largest component in a hyperbolic model of
complex networks, Annals of Applied Probability, Volume 28, Issue 1, pages 607650;
[+]
We consider the component structure of a recent model of random graphs on the
hyperbolic plane that was introduced by Krioukov et al. The model exhibits a
power law degree sequence, small distances and clustering, features that are
associated with the socalled complex networks. The model is controlled by two
parameters \(\alpha\) and \(\nu\) where, roughly speaking, \(\alpha\) controls the
exponent of the power law and \(\nu\) controls the average degree. Refining
earlier results, we are able to show a law of large numbers for the largest
component. That is, we show that the fraction of points in the largest
component tends in probability to a constant \(c\) that depends only on
\(\alpha,\nu\), while all other components are sublinear. We also study how \(c\)
depends on \(\alpha, \nu\). To deduce our results, we introduce a local
approximation of the random graph by a continuum percolation model on
\(\mathbb{R}^2\) that may be of independent interest.

P. Heinig, T. Müller, M. Noy, A. Taraz,
Logical limit laws for minorclosed classes of graphs, Journal of Combinatorial Theory series B, Volume 130, pages 158206.
(A supporting document for this paper can be found here.)
[+]
Let \(\cal G\) be an addable, minorclosed class of graphs. We prove that the zeroone law holds in monadic secondorder logic (MSO) for the random graph drawn uniformly at random from all connected graphs in \(G\) on \(n\) vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in \(G\) on \(n\) vertices. We also prove analogues of these results for the class of graphs embeddable on a fixed surface, provided we restrict attention to first order logic (FO). Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface \(S\). We also prove that the closure of the set of limiting probabilities is always the finite union of at least two disjoint intervals, and that it is the same for FO and MSO. For the classes of forests and planar graphs we are able to determine the closure of the set of limiting probabilities precisely. For planar graphs it consists of exactly 108 intervals, each of length \(\approx 5 · 10^{−6}\). Finally, we analyse examples of nonaddable classes where the behaviour is quite different. For instance, the zeroone law does not hold for the random caterpillar on \(n\) vertices, even in FO.

T. Müller, M. Noy,
The first order convergence law fails for random perfect graphs, Random Structures and Algorithms, Volume 53, Issue 4, pages 717727;
[+]
We consider first order expressible properties of random perfect graphs. That is, we pick a graph \(G_n\) uniformly at random from all
(labelled) perfect graphs on \(n\) vertices and consider the probability that it satisfies some graph property that can be expressed
in the first order language of graphs.
We show that there exists such a first order expressible property for which the probability that \(G_n\) satisfies it does not
converge as \(n\to\infty\).

T. Müller, M. Staps,
The diameter of KPKVB random graphs, Advances in Applied Probability, Volume 51, Issue 2, Pages 358–377;
[+]
We consider a model for complex networks that was recently proposed as a model for complex networks by Krioukov et al.
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 powerlaw degree distribution 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 \(\alpha, \nu > 0\) which we think of
as constant. Roughly speaking \(\alpha\) controls the power law exponent of the degree sequence and \(\nu\) the average degree.
Earlier work of Kiwi and Mitsche has shown that when \(\alpha < 1\) (which corresponds to the exponent of the power law degree sequence
being \(< 3\)) then the diameter of the largest component is a.a.s. polylogarithmic
in \(N\). Friedrich and Krohmer have shown it is a.a.s. \(\Omega(\log N)\) and they improved the exponent of the polynomial in \(\log N\) in the upper bound.
Here we show the maximum diameter over all components is a.a.s. \(O(\log N)\) thus giving a bound that is tight up to a multiplicative constant.

T. Müller, M. Stehlík,
Generalised Mycielski graphs and the BorsukUlam theorem, Electronic Journal of Combinatorics, Volume 26, Issue 4, 7 pages;
[+]
Stiebitz determined the chromatic number of generalised Mycielski graphs using the topological method of Lovasz, which invokes the BorsukUlam theorem. Van Ngoc and Tuza used elementary combinatorial arguments to prove Stiebitz's theorem for 4chromatic generalised Mycielski graphs, and asked if there is also an elementary combinatorial proof for higher chromatic number. We answer their question by showing that Stiebitz's theorem can be deduced from a version of Fan's combinatorial lemma. Our proof uses topological terminology, but is otherwise completely discrete and could be rewritten to avoid topology altogether. However, doing so would be somewhat artificial, because we also show that Stiebitz's theorem is equivalent to the BorsukUlam theorem.

T. Müller, M. D. Penrose,
Optimal Cheeger cuts and bisections of random geometric graphs, Annals of Applied Probability, Volume 30, Issue 3, 14581483;
[+]
The Cheeger constant of a graph is the minimum surfacetovolume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and volume here: the simplest method is to count boundary edges (for the surface) and vertices (for the volume). We show that for a geometric (possibly weighted) graph on n random points in a \(d\)dimensional domain with Lipschitz boundary and with distance parameter decaying more slowly (as a function of \(n\)) than the connectivity threshold, the Cheeger constant (under several possible definitions of surface and volume), also known as conductance, suitably rescaled, converges for large n to an analogous Cheegertype constant of the domain. Previously, García Trillos et al. had shown this for \(d\geq 3\) but had required an extra condition on the distance parameter when \(d=2\).

N. Fountoulakis, D. Mitsche, T. Müller, M. Schepers,
Hamilton cycles and perfect matchings in the KPKVB model, Stochastic Processes and Applications, Volume 131, pages 340–358;
[+]
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 powerlaw degree distribution, “short distances” and a nonvanishing clustering coefficient. The model is specified using three parameters: the number of nodes n, which we think of as going to infinity, and \(\alpha\), \(\nu\) > 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 \(\alpha < 1/2\) and \(\nu = \nu(\alpha)\) sufficiently small, the model does not contain a perfect matching with high probability, whereas for every \(\alpha < 1/2\) and \(\nu = \nu(\alpha)\) sufficiently large, the model contains a Hamilton cycle with high probability.

N. Fountoulakis, W.L.F. van der Hoorn, T. Müller, M. Schepers,
Clustering in a hyperbolic model of complex networks, Electronic Journal of Probability, Vol. 26, paper no. 13, pages 1132;
[+]
In this paper we consider the clustering coefficient, and clustering function 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 powerlaw degree distribution, ``short distances'' and a nonvanishing clustering coefficient. The model is specified using three parameters: the number of nodes \(n\), which we think of as going to infinity, and \(\alpha, \nu > 0\), which we think of as constant. Roughly speaking, the parameter \(\alpha\) controls the power law exponent of the degree sequence and \(\nu\) the average degree.
Here we show that the clustering coefficient tends in probability to a constant \(\gamma\) that we give explicitly as a closed form expression in terms of \(\alpha, \nu\) and certain special functions. This improves earlier work by Gugelmann et al., who proved that the clustering coefficient remains bounded away from zero with high probability, but left open the issue of convergence to a limiting constant. Similarly, we are able to show that \(c(k)\), the average clustering coefficient over all vertices of degree exactly \(k\), tends in probability to a limit \(\gamma(k)\) which we give explicitly as a closed form expression in terms of \(\alpha, \nu\) and certain special functions. We are able to extend this last result also to sequences \((k_n)_n\) where \(k_n\) grows as a function of \(n\).
Our results show that \(\gamma(k)\) scales differently, as \(k\) grows, for different ranges of \(\alpha\). More precisely, there exists constants \(c_{\alpha,\nu}\) depending on \(\alpha\) and \(\nu\), such that as \(k \to \infty\), \(\gamma(k) \sim c_{\alpha,\nu} \cdot k^{2  4\alpha}\) if \(\frac{1}{2} < \alpha < \frac{3}{4}\), \(\gamma(k) \sim c_{\alpha,\nu} \cdot \log(k) \cdot k^{1}\) if \(\alpha=\frac{3}{4}\) and \(\gamma(k) \sim c_{\alpha,\nu} \cdot k^{1}\) when \(\alpha > \frac{3}{4}\).
These results contradict a claim of Krioukov et al., which stated that the limiting values \(\gamma(k)\) should always scale with \(k^{1}\) as we let \(k\) grow.

B.T. Hansen, T. Müller,
The critical probability for Voronoi percolation in the hyperbolic plane tends to 1/2, Random Structures and Algorithms, Volume 60, Issue 1, pages 5467;
[+]
We consider percolation on the Voronoi tessellation generated by a homogeneous
Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster tends to 1/2 as the intensity of the Poisson process tends to infinity. This confirms a conjecture of Benjamini and Schramm.

A. Larrauri, T. Müller, M. Noy,
Limiting probabilities of first order properties of random sparse graphs and hypergraphs, Random Structures Algorithms, Volume 60, Issue 3, pages 506–526;
[+]
Let \(G_n\) be the binomial random graph \(G(n,p=c/n)\) in the sparse regime, which as is wellknown undergoes a phase transition at \(c=1\).
Lynch (Random Structures Algorithms, 1992) showed that for every first order sentence
\(\phi\), the limiting probability that \(G_n\) satisfies \(\phi\) as \(n\to\infty\) exists, and moreover it is an analytic function of \(c\).
In this paper we consider the closure \(\overline{L_c}\) in \([0,1]\) of the set \(L_c\) of all limiting probabilities of first order sentences in \(G_n\). We show that there exists a critical value \(c_0\approx0.93\) such that \(\overline{L_c}=[0,1]\) when
\(c\geq c_0\), whereas \(\overline{L_c}\) misses at least one subinterval when \({c < c_0}\).
We extend these results to random \(d\)uniform sparse hypergraphs, where the probability of a hyperedge is given by \(p=c/n^{d−1}\).

R. Hod, M. Krivelevich, T. Müller, A. Naor, N. Wormald,
Component games on random graphs, Combinatorica, Volume 42, Issue Suppl 1, pages 11891229;
[+]
In the \((1 : b)\) component game played on a graph \(G\), two players, Maker and Breaker, alternately claim 1 and \(b\) previously unclaimed edges of \(G\), respectively. Maker's aim is to maximise the size of a largest connected component in her graph, while Breaker is trying to minimise it. We show that the outcome of the game on the binomial random graph is strongly correlated with the appearance of a nonempty
\((b + 2)\)core in the graph.
For any integer \(k\), the \(k\)core of a graph is its largest subgraph of minimum degree at least \(k\). Pittel, Spencer and Wormald showed in 1996 that for any \(k \geq 3\) there exists an explicitly dened constant \(c_k\) such that \(p = c_k/n\)
is the threshold function for the appearance of the \(k\)core in \(G \sim G (n, p)\). More precisely, \(G \sim G (n, c/n)\) has WHP a linearsize \(k\)core for constant \(c > c_k\), and an empty \(k\)core when \(c < c_k\).
We show that for any positive constant integer b, when playing the \((1 : b)\) component game on \(G \sim G (n, c/n)\), Maker can WHP build a linearsize component if \(c > c_{b+2}\), while Breaker can WHP prevent Maker from building larger than polylogarithmicsize components if \(c < c_{b+2}\).
For Breaker's strategy, we prove a theorem which may be of independent interest. The standard algorithm for computing the kcore of any graph is to repeatedly delete (peel) all vertices of degree less than k, as long as such vertices remain. When
\(G \sim G(n,c/n)\) for \(c < c_k\) , it was shown by Jiang, Mitzenmacher and Thaler that \(\log_{k−1}\log n + \Theta(1)\) peeling iterations are WHP necessary and sucient to obtain the (empty) \(k\)core of \(G\). Our theorem states that already after a constant number of iterations, \(G\) is WHP shattered into pieces of polylogarithmic size.

T. Müller, M. Stehlík,
On the contractibility of random VietorisRips complexes, Discrete and Computational Geometry, Volume 69, Issue 4, pages 1139–1156;
[+]
We show that the VietorisRips complex \({\mathcal R}(n,r)\) built over \(n\) points sampled at random from a uniformly positive probability
measure on a convex body \(K\subseteq {\mathbb R}^d\) is a.a.s. contractible when \(r \geq c \left(\frac{\ln n}{n}\right)^{1/d}\)
for a certain constant that depends on \(K\) and the probability measure used. This answers a question of Kahle.
We also extend the proof to show that if \(K\) is a compact, smooth \(d\)manifold with boundary  but not necessarily convex  then
\({\mathcal R}(n,r)\) is a.a.s. homotopy equivalent to \(K\) when \(c_1 \left(\frac{\ln n}{n}\right)^{1/d} \leq r \leq c_2\)
for constants \(c_1=c_1(K), c_2=c_2(K)\).
Our proofs expose a connection with the game of cops and robbers.

J. He, T. Müller, T.W. Verstraaten,
Cycles in Mallows random permutations, Random Structures and Algorithms, Volume 63, Issue 4, pages 10541099;
[+]
We study cycle counts in permutations of \(1,\dots,n\) drawn at random according to the Mallows distribution.
Under this distribution, each permutation \(\pi \in S_n\) is selected with probability proportional to
\(q^{\text{inv}(\pi)}\), where \(q>0\) is a parameter and \(\text{inv}(\pi)\) denotes the number of inversions of \(\pi\).
For \(\ell\) fixed, we study the vector \((C_1(\Pi_n),\dots,C_\ell(\Pi_n))\) where \(C_i(\pi)\) denotes the
number of cycles of length \(i\) in \(\pi\) and \(\Pi_n\) is sampled according to the Mallows distribution.
When \(q=1\) the Mallows distribution simply samples a permutation
of \(1,\dots,n\) uniformly at random.
A classical result going back to Kolchin and Goncharoff states that in this case, the vector
of cycle counts tends in distribution to a vector of independent Poisson random variables, with means
\(1,\frac12,\frac13,\dots,\frac{1}{\ell}\).
Here we show that if \({0 < q < 1}\) is fixed and \(n\to\infty\) then there are positive constants \(m_i\) such that each \(C_i(\Pi_n)\) has mean
\( (1+o(1)) \cdot m_i\cdot n \) and the vector of cycle counts can be suitably rescaled to tend to a joint Gaussian distribution.
Our results also show that when \(q>1\) there is a striking difference between the behaviour of the even and the odd cycles.
The even cycle counts still have linear means, and when properly rescaled tend to a multivariate Gaussian distribution.
For the odd cycle counts on the other hand, the limiting behaviour depends on the parity of \(n\) when \(q>1\).
Both \((C_1(\Pi_{2n}),C_3(\Pi_{2n}),\dots)\) and
\((C_1(\Pi_{2n+1}),C_3(\Pi_{2n+1}),\dots)\) have discrete limiting distributions  they do not
need to be renormalized  but the two limiting distributions are distinct for all \(q>1\).
We describe these limiting distributions in terms of Gnedin and Olshanski's biinfinite extension of the Mallows model.
We investigate these limiting distributions further, and study the behaviour of the constants involved in the Gaussian limit laws.
We for example show that as \(q\downarrow 1\) the expected number of 1cycles tends to \(1/2\)  which, curiously, differs
from the value corresponding to \(q=1\).
In addition we exhibit an interesting ``oscillating'' behaviour in the limiting probability measures
for \(q>1\) and \(n\) odd versus \(n\) even.

B.T. Hansen, T. Müller,
PoissonVoronoi percolation in the hyperbolic plane with small intensities, submitted, 70 pages.
[+]
We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane.
We show that the critical probability for the existence of an infinite cluster is asymptotically equal to \(\pi \lambda/3\) as \(\lambda\to0\).
This answers a question of Benjamini and Schramm.

T. Müller, F. Skerman, T.W. Verstraaten,
Logical limit laws for Mallows random permutations, submitted, 53 pages.
[+]
A random permutation \(\Pi_n\) of \(\{1,\dots,n\}\) follows the \(\text{Mallows}(n,q)\) distribution with parameter \(q>0\) if \({\mathbb P}\left( \Pi_n = \pi \right)\) is proportional to \(q^{\text{inv}(\pi)}\) for all \(\pi\).
Here \(\text{inv}(\pi)\) \(= \{ i < j : \pi(i)> \pi(j) \}\)
denotes the number of inversions of \(\pi\).
We consider properties of permutations that can be expressed by the sentences of two different logical languages.
Namely, the theory of one bijection (\(\text{TOOB}\)), which describes
permutations via a single binary relation, and the theory of two orders (\(\text{TOTO}\)),
where we describe permutations by two total orders.
We say that the convergence law holds with respect to
one of these languages if, for every sentence \(\phi\) in the language, the probability \({\mathbb P} (\Pi_n\text{ satisfies } \phi)\)
converges to a limit as \(n\to\infty\). If moreover that limit is \(\in\) \(\{0,1\}\) for all sentences, then the zeroone law holds.
We will show that with respect to \(\text{TOOB}\) the \(\text{Mallows}(n,q)\) distribution satisfies the convergence
law but not the
zeroone law when \(q=1\), when \( 0 < q < 1 \) is fixed the zeroone law holds, and
for fixed \(q>1\) the convergence law fails.
We will prove that with respect to \(\text{TOTO}\) the \(\text{Mallows}(n,q)\) distribution satisfies the convergence law
but not the zeroone law for any fixed \(q\neq 1\), and that if \(q=q(n)\) satisfies
\(1 1/\log^*n < q < 1 + 1/\log^*n\) then \(\text{Mallows}(n,q)\) fails the convergence law.
Here \(\log^*\) denotes the discrete inverse of the tower function.