Tomasz Łuczak

Component behavior near the critical point of the random graph process

We study the behavior of a random graph process (G(n, M)):’ ) for M(n) = n12 + s and I ~ 1 ~ n ~ + m . Among others we find the number of components in G(n, M) and estimate the number of vertices and edges in the kth largest component of G(n, M), for any natural number k, Moreover, it is shown that, with probability 1 o(l), when M ( n ) = n12 + s, s 3 C 2 + -a, then during a random graph process in some step M’ > M a “new” largest component will emerge, ,whereas when s3n-*+m, the largest component of G(n, M) remains largest until the very end of the process.

On K 4 -Free Subgraphs of Random Graphs.

For 0<γ≤1 and graphsG andH, writeG→γH if any γ-proportion of the edges ofG spans at least one copy ofH inG. As customary, writeKr for the complete graph onr vertices. We show that for every fixed real η>0 there exists a constantC=C(η) such that almost every random graphGn,p withp=p(n)≥Cn−2/5 satisfiesGn,p→2/3+ηK4. The proof makes use of a variant of Szemerédis regularity lemma for sparse graphs and is based on a certain superexponential estimate for the number of pseudo-random tripartite graphs whose triangles are not too well distributed. Related results and a general conjecture concerningH-free subgraphs of random graphs in the spirit of the Erdős-Stone theorem are discussed.

Unmodulated spin chains as universal quantum wires

We study a quantum state transfer between two qubits interacting with the ends of a quantum wire consisting of linearly arranged spins coupled by an excitation conserving, time-independent Hamiltonian. We show that, if we control the coupling between the source and the destination qubits and the ends of the wire, the evolution of the system can lead to an almost perfect transfer even in the case in which all nearest-neighbour couplings between the internal spins of the wire are equal.

Size and connectivity of the k-core of a random graph

Abstract Let G(n, p) be a graph with n labelled vertices in which each edge is present independently with probability p = p(n) and let C(k; n, p) be the maximal subgraph of G(n, p) with the minimal degree at least k = k(n). In this paper we estimate the size of C(k; n, p) and consider the probability that C(k; n, p) is k-connected when n→∞.

The phase transition in a random hypergraph

We show that in the evolution of the random d-uniform hypergraph Gd(n,M) the phase transition occurs when M = n/d(d - 1) + O(n2/3). We also prove local limit theorems for the distribution of the size of the largest component of Gd(n,M) in the subcritical and in the early supercritical phase.

R(Cn,Cn,Cn)≤(4+o(1))n

In the paper we are concerned with the asymptotic behaviour of the Ramsey number R(Cn , Cn , Cn). It has been conjectured by Bondy and Erdo s [1] (see also [2]) that R(Cn , Cn , Cn) 4n&3, which would give the correct value of R(Cn , Cn , Cn) in the case when n is odd. Although we are not able to settle this problem, we show that the value of R(Cn , Cn , Cn) does not grow with n much faster than 4n.

Ramsey properties of random graphs

Abstract Let F → ( G ) r v [ F → ( G ) r e ] mean that for every r -coloring of the vertices [edges] of graph F there is a monochromatic copy of G in F . A rational d is said to be crucial for property A if for some constants c and C the probability that the binomial random graph K ( n , p ) has A tends to 0 when np d c and tends to 1 while np d > C , p = p ( n ), n → ∞. Let | G | and e ( G ) stand for the number of the vertices and edges of a graph G , respectively. We prove that max H⊆G (e (H) |H| − 1) is crucial for K ( n , p ) → ( G ) r v , whereas 2 is crucial for K ( n , p ) → ( K 3 ) 2 e . The existence of sparse Ramsey graphs is also deduced.

On the independence and chromatic numbers of random regular graphs

Abstract Let Gr denote a random r-regular graph with vertex set {1, 2, …, n} and α(Gr) and χ(Gr) denote respectively its independence and chromatic numbers. We show that with probability going to 1 as n → ∞ respectively |δ(G r ) − 2n r ( log r − log log r + 1 − log 2)|⩽ γn r and |χ(G r ) − r 2 log r − 8r log log r ( log ) 2 | ⩽ 8r log log r ( log r ) 2 provided r = o(nθ), θ 1 3 , 0

Biased Positional Games for Which Random Strategies are Nearly Optimal

G and natural numbers n and q let G(G; n, q) be the game on the complete graph in which two players, Maker and Breaker, alternately claim 1 and q edges respectively. Makers aim is to build a copy of G while Breaker tries to prevent it. Let . It is shown that there exist constants and such that Maker has a winning strategy in G(G; n, q) if , while for the game can be won by Breaker.

On Random Generation of the Symmetric Group

We prove that the probability i ( n, k ) that a random permutation of an n element set has an invariant subset of precisely k elements decreases as a power of k , for k ≤ n /2. Using this fact, we prove that the fraction of elements of S n belong to transitive subgroups other than S n or A n tends to 0 when n → ∞, as conjectured by Cameron. Finally, we show that for every ∈ > 0 there exists a constant C such that C elements of the symmetric group S n , chosen randomly and independently, generate invariably S n with probability at least 1 − ∈. This confirms a conjecture of McKay.