arXiv subjects
Publications and source records attributed to Tomasz Łuczak.
For a graph $G$ the random $n$-lift of $G$ is obtained by replacing each of its vertices by a set of $n$ vertices, and joining a pair of sets by a random matching whenever the corresponding vertices of $G$ are adjacent. We show that asymptotically almost surely the random lift of a graph $G$ is hamiltonian, provided $G$ has the minimum degree at least $5$ and contains two disjoint Hamiltonian cycles whose union is not a bipartite graph.
The random triangular group Γ(n,t) is a group given by a presentation P= , where S is a set of n generators and R is a random set of t cyclically reduced words of length three. The asymptotic behavior of Γ(n,t) is in some respects similar to that of widely studied density random group introduced by Gromov. In particular, it is known that if t <= n^{3/2-ε} for some ε> 0, then with probability 1-o(1) Γ(n,t) is infinite and hyperbolic, while for t >= n^{3/2+ε}, with probability 1-o(1) it is trivial. In this note we show that Γ(n,t) collapses provided only that t <= C n^{3/2} for some constant C>0.
Let Γ(n,p) denote the binomial model of a random triangular group. We show that there exist constants c, C > 0 such that if p <= c/n^2, then a.a.s. Γ(n,p) is free and if p >= C log n/n^2 then a.a.s. Γ(n,p) has Kazhdan's property (T). Furthermore, we show that there exist constants C',c' > 0 such that if C'/n^2 <= p <= c' log n/n^2, then a.a.s. Γ(n,p) is neither free nor has Kazhdan's property (T).
Bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of "activation" on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least $r\geq2$ active neighbors become active as well. We study the size $A^*$ of the final active set. The parameters of the model are, besides $r$ (fixed) and $n$ (tending to $\infty$), the size $a=a(n)$ of the initially active set and the probability $p=p(n)$ of the edges in the graph. We show that the model exhibits a sharp phase transition: depending on the parameters of the model, the final size of activation with a high probability is either $n-o(n)$ or it is $o(n)$. We provide a complete description of the phase diagram on the space of the parameters of the model. In particular, we find the phase transition and compute the asymptotics (in probability) for $A^*$; we also prove a central limit theorem for $A^*$ in some ranges. Furthermore, we provide the asymptotics for the number of steps until the process stops.
Let $P(n,M)$ be a graph chosen uniformly at random from the family of all labeled planar graphs with $n$ vertices and $M$ edges. In the paper we study the component structure of $P(n,M)$. Combining counting arguments with analytic techniques, we show that there are two critical periods in the evolution of $P(n,M)$. The first one, of width $Θ(n^{2/3})$, is analogous to the phase transition observed in the standard random graph models and takes place for $M=n/2+O(n^{2/3})$, when the largest complex component is formed. Then, for $M=n+O(n^{3/5})$, when the complex components cover nearly all vertices, the second critical period of width $n^{3/5}$ occurs. Starting from that moment increasing of $M$ mostly affects the density of the complex components, not its size.
For a graph $L$ and an integer $k\geq 2$, $R_k(L)$ denotes the smallest integer $N$ for which for any edge-colouring of the complete graph $K_N$ by $k$ colours there exists a colour $i$ for which the corresponding colour class contains $L$ as a subgraph. Bondy and Erdős conjectured that for an odd cycle $C_n$ on $n$ vertices, $$R_k(C_n) = 2^{k-1}(n-1)+1 \text{for $n>3$.}$$ They proved the case when $k=2$ and also provided an upper bound $R_k(C_n)\leq (k+2)!n$. Recently, this conjecture has been verified for $k=3$ if $n$ is large. In this note, we prove that for every integer $k\geq 4$, $$R_k(C_n)\leq k2^kn+o(n), \text{as $n\to\infty$.}$$ When $n$ is even, Yongqi, Yuansheng, Feng, and Bingxi gave a construction, showing that $R_k(C_n)\geq (k-1)n-2k+4.$ Here we prove that if $n$ is even, then $$R_k(C_n)\leq kn+o(n), \text{as $n\to\infty$.}$$
The Vapnik-Červonenkis dimension is a complexity measure of set-systems, or hypergraphs. Its application to graphs is usually done by considering the sets of neighborhoods of the vertices (cf. Alon et al. (2006) and Chepoi, Estellon, and Vaxes (2007)), hence providing a set-system. But the graph structure is lost in the process. The aim of this paper is to introduce the notion of paired VC-dimension, a generalization of VC-dimension to set-systems endowed with a graph structure, hence a collection of pairs of subsets. The classical VC-theory is generally used in combinatorics to bound the transversality of a hypergraph in terms of its fractional transversality and its VC-dimension. Similarly, we bound the chromatic number in terms of fractional transversality and paired VC-dimension. This approach turns out to be very useful for a class of problems raised by Erdős and Simonovits (1973) asking for H-free graphs with minimum degree at least cn and arbitrarily high chromatic number, where H is a fixed graph and c a positive constant. We show how the usual VC-dimension gives a short proof of the fact that triangle-free graphs with minimum degree at least n/3 have bounded chromatic number, where $n$ is the number of vertices. Using paired VC-dimension, we prove that if the chromatic number of $H$-free graphs with minimum degree at least cn is unbounded for some positive c, then it is unbounded for all c<1/3. In other words, one can find H-free graphs with unbounded chromatic number and minimum degree arbitrarily close to n/3. These H-free graphs are derived from a construction of Hajnal. The large chromatic number follows from the Borsuk-Ulam Theorem.
We study the size of the largest clique $ω(G(n,α))$ in a random graph $G(n,α)$ on $n$ vertices which has power-law degree distribution with exponent $α$. We show that for `flat' degree sequences with $α>2$ whp the largest clique in $G(n,α)$ is of a constant size, while for the heavy tail distribution, when $0<α<2$, $ω(G(n,α))$ grows as a power of $n$. Moreover, we show that a natural simple algorithm whp finds in $G(n,α)$ a large clique of size $(1+o(1))ω(G(n,α))$ in polynomial time.
For a sequence p=(p(1),p(2), ...) let G(n,p) denote the random graph with vertex set {1,2, ...,n} in which two vertices i, j are adjacent with probability p(|i-j|), independently for each pair. We study how the convergence of probabilities of first order properties of G(n,p), can be affected by the behaviour of p and the strength of the language we use.
Limiting distributions are derived for the sparse connected components that are present when a random graph on $n$ vertices has approximately $\half n$ edges. In particular, we show that such a graph consists entirely of trees, unicyclic components, and bicyclic components with probability approaching $\sqrt{2\over 3} \cosh\sqrt{5\over 18}\approx0.9325$ as $n\to\infty$. The limiting probability that it consists of trees, unicyclic components, and at most one other component is approximately 0.9957; the limiting probability that it is planar lies between 0.987 and 0.9998. When a random graph evolves and the number of edges passes $\half n$, its components grow in cyclic complexity according to an interesting Markov process whose asymptotic structure is derived. The probability that there never is more than a single component with more edges than vertices, throughout the evolution, approaches $5π/18\approx0.8727$. A ``uniform'' model of random graphs, which allows self-loops and multiple edges, is shown to lead to formulas that are substantially simpler than the analogous formulas for the classical random graphs of Erdős and Rényi. The notions of ``excess'' and ``deficiency,'' which are significant characteristics of the generating function as well as of the graphs themselves, lead to a mathematically attractive structural theory for the uniform model. A general approach to the study of stopping configurations makes it possible to sharpen previously obtained estimates in a uniform manner and often to obtain closed forms for the constants of interest. Empirical results are presented to complement the analysis, indicating the typical behavior when $n$ is near 20000.