The critical random graph,with martingales

We give a short proof that the largest component C ₁ of the random graph G(n, 1/n) is of size approximately n ^2/3. The proof gives explicit bounds for the probability that the ratio is very large or very small. In particular, the probability that n ^−2/3|C ₁| exceeds A is at most e ^{- cA3}{e^{ - c{A^3}}} for some c > 0.     

Random Subgraphs Of Finite Graphs: III. The Phase Transition For The n-Cube

We study random subgraphs of the n-cube {0,1}ⁿ, where nearest-neighbor edges are occupied with probability p. Let p_c(n) be the value of p for which the expected size of the component containing a fixed vertex attains the value λ2^n/3, where λ is a small positive constant. Let ε=n(p−p_c(n)). In two previous papers, we showed that the largest component inside a scaling window given by |ε|=Θ(2^−n/3) is of size Θ(2^2n/3), below this scaling window it is at most 2(log 2)nε⁻², and above this scaling window it is at most O(ε2ⁿ). In this paper, we prove that for the size of the largest component is at least Θ(ε2ⁿ), which is of the same order as the upper bound. The proof is based on a method that has come to be known as “sprinkling,” and relies heavily on the specific geometry of the n-cube.     

Growth and recurrence of stationary random walks

 Let (X _n ,n≥1) be a real-valued ergodic stationary stochastic process, and let (Y _n =X ₁ +…+X _n ,n≥1) be the associated random walk. We prove the following: if the sequence of distributions of the random variables Y _n /n,n≥1, is uniformly tight (or, more generally, does not have the zero measure as a vague limit point), then there exists a real number c such that the random walk (Y _n −nc,n≥1) is recurrent. If this sequence of distributions converges to a probability measure ρ on ℝ (or, more generally, has a nonzero limit ρ in the vague topology), then (Y _n −nc,n≥1) is recurrent for ρ−a.e.cℝ. Received: 24 September 2001 / Revised version: 1 August 2002 / Published online: 24 October 2002 The first author was partially supported by the FWF research project P14379-MAT. Mathematics Subject Classification (2000): 37A20, 37A50, 60G10, 60G50 Key words or phrases: Recurrent stationary random walks – Recurrent cocycles     

Merging percolation on Zd and classical random graphs: Phase transition

We study a random graph model which is a superposition of bond percolation on Z^d with parameter p, and a classical random graph G(n,c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so‐called “rank 1 case” of inhomogeneous random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters c ≥ 0 and 0 ≤ p < p_c, where p_c = p_c(d) is the critical probability for the bond percolation on Z^d. The phase transition is of second order as in the classical random graph. We find the scaled size of the largest connected component in the supercritical regime. We also provide a sharp upper bound for the largest connected component in the subcritical regime. The latter is a new result for inhomogeneous random graphs with unbounded kernels. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Correlated equilibria of games with many players

Let G _m,n be the class of strategic games with n players, where each player has m≥2 pure strategies. We are interested in the structure of the set of correlated equilibria of games in G _m,n when n→∞. As the number of equilibrium constraints grows slower than the number of pure strategy profiles, it might be conjectured that the set of correlated equilibria becomes large. In this paper, we show that (1) the average relative measure of the set of correlated equilibria is smaller than 2⁻ⁿ; and (2) for each 1<c<m, the solution set contains c ⁿ correlated equilibria having disjoint supports with a probability going to 1 as n grows large. The proof of the second result hinges on the following inequality: Let c ₁, …, c _l be independent and symmetric random vectors in R ^k, l≥k. Then the probability that the convex hull of c ₁, …, c _l intersects R ^k ₊ is greater than or equal to . Received: December 1998/Final version: March 2000     

On factors in random graphs

The following result is proved: LetG _n,p be a random graph withn vertices and probabilityp for an edge. Ifp is such that the random graph has min-degree at leastr with probability 1, then anyf-factor 1≦f≦r exists with probability 1, asn→∞.     

A phase transition phenomenon in a random directed acyclic graph

Let a random directed acyclic graph be defined as being obtained from the random graph G_{n, p} by orienting the edges according to the ordering of vertices. Let γ_n* be the size of the largest (reflexive, transitive) closure of a vertex. For p=c(log n)/n, we prove that, with high probability, γ_n* is asymptotic to n^c log n, 2n(log log n)/log n, and n(1−1/c) depending on whether c<1, c=1, or c>1. We also determine the limiting distribution of the first vertex closure in all three ranges of c. As an application, we show that the expected number of comparable pairs is asymptotic to n^1+c/c log n, ½(n(log log n)/log n)², and ½(n(1−1/c))², respectively. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 164–184, 2001     

Subgraphs of colour-critical graphs

Some problems and results on the distribution of subgraphs in colour-critical graphs are discussed. In section 3 arbitrarily largek-critical graphs withn vertices are constructed such that, in order to reduce the chromatic number tok−2, at leastc _k n ² edges must be removed. In section 4 it is proved that a 4-critical graph withn vertices contains at mostn triangles. Further it is proved that ak-critical graph which is not a complete graph contains a (k−1)-critical graph which is not a complete graph.     

Random subgraphs of finite graphs: I. The scaling window under the triangle condition

We study random subgraphs of an arbitrary finite connected transitive graph ?? obtained by independently deleting edges with probability 1 ? p. Let V be the number of vertices in ??, and let Ω be their degree. We define the critical threshold p_c = p_c (??, λ) to be the value of p for which the expected cluster size of a fixed vertex attains the value λV^1/3, where λ is fixed and positive. We show that, for any such model, there is a phase transition at p_c analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold p_c. In particular, we show that the largest cluster inside a scaling window of size |p ? p_c| = Θ(Ω^?1V^?1/3) is of size Θ(V^2/3), while, below this scaling window, it is much smaller, of order O(?^?2 log(V?³)), with ? = Ω(p_c ? p). We also obtain an upper bound O(Ω(p ? p_c)V) for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order Θ(Ω(p ? p_c)). Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the n‐cube and certain Hamming cubes, as well as the spread‐out n‐dimensional torus for n > 6. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

The ring lemma in three dimensions

Suppose that n cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc. The sharp ring lemma in two dimensions states that no disc has a radius below c _n (R ²) = (F _2n−3−1)⁻¹—where F _k denotes the kth Fibonacci number—and that the lower bound is attained in essentially unique Apollonian configurations. In this article, generalizations of the ring lemma to three dimensions are discussed, a version of the ring lemma in three dimensions is proved, and a natural generalization of the extremal two-dimensional configuration—thought to be extremal in three dimensions—is given. The sharp three-dimensional ring lemma constant of order n is shown to be bounded from below by the two-dimensional constant of order n − 1.     

The hamiltonian chromatic number of a connected graph without large hamiltonian-connected subgraphs

If G is a connected graph of order n ⩾ 1, then by a hamiltonian coloring of G we mean a mapping c of V (G) into the set of all positive integers such that |c(x) − c(y)| ⩾ n − 1 − D _G (x, y) (where D _G (x, y) denotes the length of a longest x − y path in G) for all distinct x, y ∈ V (G). Let G be a connected graph. By the hamiltonian chromatic number of G we mean

Automorphisms of random graphs with specified vertices

Conditions are found under which the expected number of automorphisms of a large random labelled graph with a given degree sequence is close to 1. These conditions involve the probability that such a graph has a given subgraph. One implication is that the probability that a random unlabelledk-regular simple graph onn vertices has only the trivial group of automorphisms is asymptotic to 1 asn → ∞ with 3≦k=O(n ^1/2−c). In combination with previously known results, this produces an asymptotic formula for the number of unlabelledk-regular simple graphs onn vertices, as well as various asymptotic results on the probable connectivity and girth of such graphs. Corresponding results for graphs with more arbitrary degree sequences are obtained. The main results apply equally well to graphs in which multiple edges and loops are permitted, and also to bicoloured graphs. Research of the second author supported by U. S. National Science Foundation Grant MCS-8101555, and by the Australian Department of Science and Technology under the Queen Elizabeth II Fellowships Scheme. Current address: Mathematics Department, University of Auckland, Auckland, New Zealand.     

Propertyf −1(S) =g −1(S) for entire and meromorphicP-adic functions

LetW be an algebraically closed filed of characteristic zero, letK be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value, and letA(K) (resp. ℳ(K)) be the set of entire (resp. meromorphic) functions inK. For everyn≥7, we show that the setS _n(b) of zeros of the polynomialx ⁿ−b (b≠0) is such that, iff, g ∈W[x] or iff, g ∈A(K), satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), thenf ⁿ=g ⁿ. For everyn≥14, we show thatS _n(b) is such that iff, g ∈W({tx}) or iff, g ∈ ℳ(K) satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), then eitherf ⁿ=g ⁿ, orfg is a constant. Analogous properties are true for complex entire and meromorphic functions withn≥8 andn≥15, respectively. For everyn≥9, we show that the setY _n(c) of zeros of the polynomial , (withc≠0 and 1) is an ursim ofn points forW[x], and forA(K). For everyn≥16, we show thatY _n(c) is an ursim ofn points forW(x), and for ℳ(K). We follow a method based on thep-adic Nevanlinna Theory and use certain improvement of a lemma obtained by Frank and Reinders.     

Cubic one-regular graphs of order twice a square-free integer

A graph is one-regular if its automorphism group acts regularly on the set of its arcs.Let n be a square-free integer.In this paper,we show that a cubic one-regular graph of order 2n exists if and only if n=3~tp1p2…p_s≥13,where t≤1,s≥1 and p_i's are distinct primes such that 3|(P_i—1). For such an integer n,there are 2~(s-1) non-isomorphic cubic one-regular graphs of order 2n,which are all Cayley graphs on the dihedral group of order 2n.As a result,no cubic one-regular graphs of order 4 times an odd square-free integer exist.     

Interior points of convex hulls

If a setX ⊂E ⁿ has non-emptyk-dimensional interior, or if some point isk-dimensional surrounded, then the classic theorem of E. Steinitz may be extended. For example ifX ⊂E ⁿ has int _k X ≠ 0, (0 ≦k≦n) and ifp ɛ int conX, thenp ɛ int conY for someY ⊂X with cardY≦2n−k+1.     

Long paths in sparse random graphs

We consider random graphs withn labelled vertices in which edges are chosen independently and with probabilityc/n. We prove that almost every random graph of this kind contains a path of length ≧(1 −α(c))n where α(c) is an exponentially decreasing function ofc. Dedicated to Tibor Gallai on his seventieth birthday     

Tiling Transitive Tournaments and Their Blow-ups

Let TT _k denote the transitive tournament on k vertices. Let TT(h,k) denote the graph obtained from TT _k by replacing each vertex with an independent set of size h≥1. The following result is proved: Let c ₂=1/2, c ₃=5/6 and c _k =1−2^{−k−log k} for k≥4. For every ∈>0 there exists N=N(∈,h,k) such that for every undirected graph G with n>N vertices and with δ(G)≥c _k n, every orientation of G contains vertex disjoint copies of TT(h,k) that cover all but at most ∈n vertices. In the cases k=2 and k=3 the result is asymptotically tight. For k≥4, c _k cannot be improved to less than 1−2^{−0.5k(1+o(1))}. This revised version was published online in June 2006 with corrections to the Cover Date.     

Weakly stable constant mean curvature hypersurfaces

Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an （n ＋ 1）-dimensional Riemannian manifold N^n＋1 whose （n - 1）th Ricci curvature satisfying Ric^N（n-1） （n - 1）c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If ｜φ｜^2 - （n- 2）√n（n- 1）｜H｜｜φ｜＋ n（2n - 1）（H^2＋ c） 〉 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c ＋ H^2 〉 0, then M must have only one end.     

On the Lower Tail Probabilities of Some Random Sequences in <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We investigate the behaviour of the logarithmic small deviation probability of a sequence (σ _n θ _n ) in l _p , 0<p≤∞, where (θ _n ) are i.i.d. random variables and (σ _n ) is a decreasing sequence of positive numbers. In particular, the example σ _n ∼n ^−μ (1+log n)^−ν is studied thoroughly. Contrary to the existing results in the literature, the rate function and the small deviation constant are expressed expli- citly in the present treatment. The restrictions on the distribution of θ ₁ are kept to an absolute minimum. In particular, the usual variance assumption is removed. As an example, the results are applied to stable and Gamma-distributed random variables.     

Groups of typeF a,b,−c

For integersa, b andc, the groupF ^a,b,−c is defined to be the group 〈R, S : R ²=RS ^aRS^bRS^−c=1〉. In this paper we identify certain subgroups of the group of affine linear transformations of finite fields of orderp ⁿ (for certainp andn) as groups of typeF ^a,b,−c for certain (not unique) choices ofa, b andc.