Packing and covering constants for certain families of trees. I

If ? denotes a family of rooted trees, let p_k(n) and c_k(n) denote the average value of the k-packing and k-covering numbers of trees in ? that have n nodes. We assume, among other things, that the generating function y of trees in ? satisfies a relation of the type y = x?(y) for some power series ?. We show that the limits of p_k(n)/n and c_k(n)/n as n → ∞ exist and we describe how to evaluate these limits.     

Disjoint bases in a polymatroid

Let f : 2^N → ⁺ be a polymatroid (an integer‐valued non‐decreasing submodular set function with f(??) = 0). We call S ? N a base if f(S) = f(N). We consider the problem of finding a maximum number of disjoint bases; we denote by m* be this base packing number. A simple upper bound on m* is given by k* = max{k : Σ_iεNf_A(i) ≥ kf_A(N),?A ? N} where f_A(S) = f(A ∪ S) ‐ f(A). This upper bound is a natural generalization of the bound for matroids where it is known that m* = k*. For polymatroids, we prove that m* ≥ (1 ? o(1))k*/lnf(N) and give a randomized polynomial time algorithm to find (1 ? o(1))k*/lnf(N) disjoint bases, assuming an oracle for f. We also derandomize the algorithm using minwise independent permutations and give a deterministic algorithm that finds (1 ? ε)k*/lnf(N) disjoint bases. The bound we obtain is almost tight because it is known there are polymatroids for which m* < (1 + o(1))k*/lnf(N). Moreover it is known that unless NP ? DTIME(n^{log log n}), for any ε > 0, there is no polynomial time algorithm to obtain a (1 + ε)/lnf(N)‐approximation to m*. Our result generalizes and unifies two results in the literature. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

The asymptotic number of labeled connected graphs with a given number of vertices and edges

Let c(n, q) be the number of connected labeled graphs with n vertices and q ≤ N = (2n ) edges. Let x = q/n and k = q ? n. We determine functions w_k ? 1. a(x) and φ(x) such that c(n, q) ? w_k(q^N)eⁿφ(x)+a(x) uniformly for all n and q ≥ n. If ? > 0 is fixed, n→ ∞ and 4q > (1 + ?)n log n, this formula simplifies to c(n, q) ? (N^q) exp(–ne^?2q/n). on the other hand, if k = o(n^1/2), this formula simplifies to c(n, n + k) ? 1/2 w_k (3/π)^1/2 (e/12k)^k/2nⁿ?^(3k?1)/2.     

Corrigendum: The cover time of the giant component of a random graph,Random Structures and Algorithms 32 (2008), 401–439

The above paper gives an asymptotically precise estimate of the cover time of the giant component of a sparse random graph. The proof as it stands is not tight enough, and in particular, Eq. (64) is not strong enough to prove (65). The o(1) term in (64) needs to be improved to o(1/(lnn)²) for (65) to follow. The following section, which replaces Section 3.6, amends this oversight. The notation and definitions are from the paper. A further correction is needed. Property P2 claims that the conductance of the giant is whp , Ω(1/lnn). The proof of P2 in the appendix of the paper is not quite complete. A complete proof of Property P2 can be found in 1 , 2 . © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Critical percolation on random regular graphs

The behavior of the random graph G(n,p) around the critical probability p_c = is well understood. When p = (1 + O(n^1/3))p_c the components are roughly of size n^2/3 and converge, when scaled by n^?2/3, to excursion lengths of a Brownian motion with parabolic drift. In particular, in this regime, they are not concentrated. When p = (1 ‐ ?(n))p_c with ?(n)n^1/3 →∞ (the subcritical regime) the largest component is concentrated around 2?^?2 log(?³n). When p = (1 + ?(n))p_c with ?(n)n^1/3 →∞ (the supercritical regime), the largest component is concentrated around 2?n and a duality principle holds: other component sizes are distributed as in the subcritical regime. Itai Benjamini asked whether the same phenomenon occurs in a random d‐regular graph. Some results in this direction were obtained by (Pittel, Ann probab 36 (2008) 1359–1389). In this work, we give a complete affirmative answer, showing that the same limiting behavior (with suitable d dependent factors in the non‐critical regimes) extends to random d‐regular graphs. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Equitable coloring of random graphs

An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most one. The least positive integer k for which there exists an equitable coloring of a graph G with k colors is said to be the equitable chromatic number of G and is denoted by χ₌(G). The least positive integer k such that for any k′ ≥ k there exists an equitable coloring of a graph G with k′ colors is said to be the equitable chromatic threshold of G and is denoted by χ₌*(G). In this paper, we investigate the asymptotic behavior of these coloring parameters in the probability space G(n,p) of random graphs. We prove that if n^?1/5+? < p < 0.99 for some 0 < ?, then almost surely χ(G(n,p)) ≤ χ₌(G(n,p)) = (1 + o(1))χ(G(n,p)) holds (where χ(G(n,p)) is the ordinary chromatic number of G(n,p)). We also show that there exists a constant C such that if C/n < p < 0.99, then almost surely χ(G(n,p)) ≤ χ₌(G(n,p)) ≤ (2 + o(1))χ(G(n,p)). Concerning the equitable chromatic threshold, we prove that if n^?(1??) < p < 0.99 for some 0 < ?, then almost surely χ(G(n,p)) ≤ χ₌* (G(n,p)) ≤ (2 + o(1))χ(G(n,p)) holds, and if < p < 0.99 for some 0 < ?, then almost surely we have χ(G(n,p)) ≤ χ₌*(G(n,p)) = O_?(χ(G(n,p))). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Improved random graph isomorphism

Canonical labeling of a graph consists of assigning a unique label to each vertex such that the labels are invariant under isomorphism. Such a labeling can be used to solve the graph isomorphism problem. We give a simple, linear time, high probability algorithm for the canonical labeling of a G(n,p) random graph for p[ω(ln⁴n/nlnlnn),1−ω(ln⁴n/nlnlnn)]. Our result covers a gap in the range of p in which no algorithm was known to work with high probability. Together with a previous result by Bollobás, the random graph isomorphism problem can be solved efficiently for p[Θ(lnn/n),1−Θ(lnn/n)].     

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     

An additive divisor problem with a growing number of factors

Let τk(n) be the number of representations ofn as the product ofk positive factors, τ(n)=τ(n). The asymptotics of Σ _n≤x τ _k (n)τ(n+1) for 80k ¹⁰ (lnlnx)³≤lnx is shown to be uniform with respect tok. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 391–406, March, 1997. Translated by N. K. Kulman     

Automorphisms of higher‐dimensional Hadamard matrices

This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P⁽ⁿ⁾(H). We apply our ideas to the matrices P⁽ⁿ⁾(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P⁽ⁿ⁾(H) is equivalent to “P⁽ⁿ⁾(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P⁽ⁿ⁾(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P⁽ⁿ⁾(Hk) when Hk is a Sylvester Hadamard matrix of order 2^k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P⁽ⁿ⁾(H) where H is equivalent to H₂. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008     

Rank‐width of random graphs

Rank‐width of a graph G, denoted by rw (G), is a width parameter of graphs introduced by Oum and Seymour [J Combin Theory Ser B 96 (2006), 514–528]. We investigate the asymptotic behavior of rank‐width of a random graph G(n, p). We show that, asymptotically almost surely, (i) if p∈(0, 1) is a constant, then rw (G(n, p)) = ?n/3??O(1), (ii) if , then rw (G(n, p)) = ?1/3??o(n), (iii) if p = c/n and c>1, then rw (G(n, p))?rn for some r = r(c), and (iv) if p?c/n and c81, then rw (G(n, p))?2. As a corollary, we deduce that the tree‐width of G(n, p) is linear in n whenever p = c/n for each c>1, answering a question of Gao [2006]. © 2011 Wiley Periodicals, Inc. J Graph Theory.     

A self-normalized Erd s—Rényi type strong law of large numbers

The original Erd s—Rényi theorem states that max_0kn∑^k+[clogn]_i=k+1X_i/[clogn]→α(c),c>0, almost surely for i.i.d. random variables {X_n, n1} with mean zero and finite moment generating function in a neighbourhood of zero. The latter condition is also necessary for the Erd s—Rényi theorem, and the function α(c) uniquely determines the distribution function of X₁. We prove that if the normalizing constant [c log n] is replaced by the random variable ∑^k+[clogn]_i=k+1(X²_i+1), then a corresponding result remains true under assuming only the exist first moment, or that the underlying distribution is symmetric.     

Coverings of the Vertices of a Graph by Small Cycles

Given a graph G with n vertices, we call c_k(G) the minimum number of elementary cycles of length at most k necessary to cover the vertices of G. We bound c_k(G) from the minimum degree and the order of the graph.     

The critical behavior of random digraphs

We study the critical behavior of the random digraph D(n,p) for np = 1 + ε, where ε = ε(n) = o(1). We show that if ε³n →—∞, then a.a.s. D(n,p) consists of components which are either isolated vertices or directed cycles, each of size O_p(|ε|^?1). On the other hand, if ε³n →∞, then a.a.s. the structure of D(n,p) is dominated by the unique complex component of size (4 + o(1))ε²n, whereas all other components are of size O_p(ε^?1). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

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     

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     

On Strong Versions of the Central Limit Theorem

Let S_n be the sum of n i.i.d.r.v. and let 1_(-∞,x)(·) be the indicator function of the interval (-∞, x). Then the sequence 1_{(-∞, x)}(S_n/√n) does not converge for any x. Likewise the arithmetic means of this sequence converge only with probability zero. But the logarithmic means converge with probability one to the standard normal distribution Ø(x). Then for any bounded and a.e. continuous function a(y) the logarithmic means of a(S_n/√n) converge a.s. to a = ∫a(y)dØ(y). The arithmetic means of a(S_nk/√n) converge to the same limit a for all subsequences n_k = [c^k], c > 1.     

Perfect matchings in random uniform hypergraphs

Let ??_k(n, p) be the random k‐uniform hypergraph on V = [n] with edge probability p. Motivated by a theorem of Erd?s and Rényi 7 regarding when a random graph G(n, p) = ??₂(n, p) has a perfect matching, the following conjecture may be raised. (See J. Schmidt and E. Shamir 16 for a weaker version.) Conjecture. Let k|n for fixed k ≥ 3, and the expected degree d(n, p) = p(). Then (Erd?s and Rényi 7 proved this for G(n, p).) Assuming d(n, p)/n^1/2 → ∞, Schmidt and Shamir 16 were able to prove that ??_k(n, p) contains a perfect matching with probability 1 ? o(1). Frieze and Janson 8 showed that a weaker condition d(n, p)/n^1/3 → ∞ was enough. In this paper, we further weaken the condition to A condition for a similar problem about a perfect triangle packing of G(n, p) is also obtained. A perfect triangle packing of a graph is a collection of vertex disjoint triangles whose union is the entire vertex set. Improving a condition p ≥ cn^?2/3+1/15 of Krivelevich 12 , it is shown that if 3|n and p ? n^?2/3+1/18, then © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 111–132, 2003     

Coloring complete bipartite graphs from random lists

Let K_n,n be the complete bipartite graph with n vertices in each side. For each vertex draw uniformly at random a list of size k from a base set S of size s = s(n). In this paper we estimate the asymptotic probability of the existence of a proper coloring from the random lists for all fixed values of k and growing n. We show that this property exhibits a sharp threshold for k ≥ 2 and the location of the threshold is precisely s(n) = 2n for k = 2 and approximately for k ≥ 3. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

The rank of random graphs

We show that almost surely the rank of the adjacency matrix of the Erd?s‐Rényi random graph G(n,p) equals the number of nonisolated vertices for any c ln n/n ≤ p ≤ 1/2, where c is an arbitrary positive constant larger than 1/2. In particular, the adjacency matrix of the giant component (a.s.) has full rank in this range. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008