Greedy colorings of uniform hypergraphs

We give a very short proof of an Erd?s conjecture that the number of edges in a non‐2‐colorable n‐uniform hypergraph is at least f(n)2ⁿ, where f(n) goes to infinity. Originally it was solved by József Beck in 1977, showing that f(n) at least clog n. With an ingenious recoloring idea he later proved that f(n) ≥ cn^1/3+o(1). Here we prove a weaker bound on f(n), namely f(n) ≥ cn^1/4. Instead of recoloring a random coloring, we take the ground set in random order and use a greedy algorithm to color. The same technique works for getting bounds on k‐colorability. It is also possible to combine this idea with the Lovász Local Lemma, reproving some known results for sparse hypergraphs (e.g., the n‐uniform, n‐regular hypergraphs are 2‐colorable if n ≥ 8). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

A lower bound on the independence number of arbitrary hypergraphs

We present a lower bound on the independence number of arbitrary hypergraphs in terms of the degree vectors. The degree vector of a vertex v is given by d(v) = (d₁(v), d₂(v), …) where d_m(v) is the number of edges of size m containing v. We define a function f with the property that any hypergraph H = (V, E) satisfies α(H) ≥ Σ_v∈V f(d(v)). This lower bound is sharp when H is a match, and it generalizes known bounds of Caro/Wei and Caro/Tuza for ordinary graphs and uniform hypergraphs. Furthermore, an algorithm for computing independent sets of size as guaranteed by the lower bound is given. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 213–221, 1999     

Component structure of the vacant set induced by a random walk on a random graph

We consider random walks on several classes of graphs and explore the likely structure of the vacant set, i.e. the set of unvisited vertices. Let Γ(t) be the subgraph induced by the vacant set of the walk at step t. We show that for random graphs G_n,p (above the connectivity threshold) and for random regular graphs G_r,r ≥ 3, the graph Γ(t) undergoes a phase transition in the sense of the well‐known ErdJW‐RSAT1100590x.png ‐Renyi phase transition. Thus for t ≤ (1 ‐ ε)t^*, there is a unique giant component, plus components of size O(log n), and for t ≥ (1 + ε)t^* all components are of size O(log n). For G_n,p and G_r we give the value of t^*, and the size of Γ(t). For G_r, we also give the degree sequence of Γ(t), the size of the giant component (if any) of Γ(t) and the number of tree components of Γ(t) of a given size k = O(log n). We also show that for random digraphs D_n,p above the strong connectivity threshold, there is a similar directed phase transition. Thus for t ≤ (1 ‐ ε)t^*, there is a unique strongly connected giant component, plus strongly connected components of size O(log n), and for t ≥ (1 + ε)t^* all strongly connected components are of size O(log n). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

The smallest degree sum that yields potentially Kr,r-graphic sequences

We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(K_r,r,n) such that every n-term graphic sequence π = (d₁,d₂,...,d_n) with term sum σ(π) = d₁ + d₂ + ... + d_n ≥ σ(K_r,r,n) is potentially K_r,r-graphic, where K_r,r is an r × r complete bipartite graph, i.e. π has a realization G containing K_r,r as its subgraph. In this paper, the values σ(K_r,r,n) for even r and n ≥ 4r² - r - 6 and for odd r and n ≥ 4r² + 3r - 8 are determined.     

Matchings in hypergraphs of large minimum degree

It is well known that every bipartite graph with vertex classes of size n whose minimum degree is at least n/2 contains a perfect matching. We prove an analog of this result for hypergraphs. We also prove several related results that guarantee the existence of almost perfect matchings in r‐uniform hypergraphs of large minimum degree. Our bounds on the minimum degree are essentially best possible. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 269–280, 2006     

On fractional K‐factors of random graphs

Let K be a graph on r vertices and let G = (V,E) be another graph on ∣V ∣ = n vertices. Denote the set of all copies of K in G by 𝒦. A non‐negative real‐valued function f : 𝒦→ ℝ₊ is called a fractional K‐factor if ∑ _{K:v∈K∈𝒦}f(K) ≤ 1 for every v ∈ V and ∑ _K∈𝒦f(K) = n/r. For a non‐empty graph K let d(K) = e(K)/v(K) and d⁽¹⁾(K) = e(K)/(v(K) ‐ 1). We say that K is strictly K₁‐balanced if for every proper subgraph K^′ ⊊ K, d⁽¹⁾(K^′) < d⁽¹⁾(K). We say that K is imbalanced if it has a subgraph K^′ such that d(K^′) > d(K). Considering a random graph process on n vertices, we show that if K is strictly K₁‐balanced, then with probability tending to 1 as n →∞, at the first moment τ₀ when every vertex is covered by a copy of K, the graph has a fractional K‐factor. This result is the best possible. As a consequence, if K is K₁‐balanced, we derive the threshold probability function for a random graph to have a fractional K‐factor. On the other hand, we show that if K is an imbalanced graph, then for asymptotically almost every graph process there is a gap between τ₀ and the appearance of a fractional K‐factor. We also introduce and apply a criteria for perfect fractional matchings in hypergraphs in terms of expansion properties. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Coloring uniform hypergraphs with few colors

Let m(r, k) denote the minimum number of edges in an r‐uniform hypergraph that is not k‐colorable. We give a new lower bound on m(r, k) for fixed k and large r. Namely, we prove that if k ≥ 2ⁿ, then m(r, k) ≥ ?(k)k^r(r/ln r)^n/(n+1). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Randomly coloring sparse random graphs with fewer colors than the maximum degree

We analyze Markov chains for generating a random k‐coloring of a random graph G_n,d/n. When the average degree d is constant, a random graph has maximum degree Θ(log n/log log n), with high probability. We show that, with high probability, an efficient procedure can generate an almost uniformly random k‐coloring when k = Θ(log log n/log log log n), i.e., with many fewer colors than the maximum degree. Previous results hold for a more general class of graphs, but always require more colors than the maximum degree. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Weak quasi‐randomness for uniform hypergraphs

We study quasi‐random properties of k‐uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung‐Graham‐Wilson theorem for quasi‐random graphs. Moreover, let K_k be the complete graph on k vertices and M(k) the line graph of the graph of the k‐dimensional hypercube. We will show that the pair of graphs (K_k,M(k)) has the property that if the number of copies of both K_k and M(k) in another graph G are as expected in the random graph of density d, then G is quasi‐random (in the sense of the Chung‐Graham‐Wilson theorem) with density close to d. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Asymptotic packing and the random greedy algorithm

Let H be an r-uniform hypergraph satisfying deg(x) = D(1 + o(1)) for each vertex xϵ V(H) and deg(x, y) = o(D) for each pair of vertices x, y ϵ V(H), where D → infinity. Recently, J. Spencer [5] showed, using a branching process approach, that almost surely the random greedy algorithm finds a packing of size at least n/r(1 − o(1)) for this class of hypergraphs. In this paper, we show an alternative proof of this via “nibbles.” Further, let T_α be the number of edges that the random greedy algorithm has to consider before yielding a packing of size [n/r · (1 − α)]. We show that almost surely T_α ∼ (1/α)^r−1 · n/r(r − 1) as α → 0⁺ holds. © 1996 John Wiley & Sons, Inc.     

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     

Limit theory for the random on‐line nearest‐neighbor graph

In the on‐line nearest‐neighbor graph (ONG), each point after the first in a sequence of points in ?^d is joined by an edge to its nearest neighbor amongst those points that precede it in the sequence. We study the large‐sample asymptotic behavior of the total power‐weighted length of the ONG on uniform random points in (0,1)^d. In particular, for d = 1 and weight exponent α > 1/2, the limiting distribution of the centered total weight is characterized by a distributional fixed‐point equation. As an ancillary result, we give exact expressions for the expectation and variance of the standard nearest‐neighbor (directed) graph on uniform random points in the unit interval. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Degree Sequences and Chromatic Numbers of Graphs

 We prove that if G runs over the set of graphs with a fixed degree sequence d, then the values χ(G) of the function chromatic number completely cover a line segment [a,b] of positive integers. Thus for an arbitrary graphical sequence d, two invariants minχ(d):=a and maxχ(d):=b naturally arise. For a regular graphical sequence d=r ⁿ :=(r,r,…,r) where r is the degree and n is the number of vertices, the exact values of a and b are found in all situations, except the case where n and r are both even and n<2r. Received: September 16, 2000 Final version received: December 13, 2001 Acknowledgments. We would like to thank Professor Tommy R. Jensen for his useful comment and editing thorough the paper.     

A (1 + ϵ)‐approximation algorithm for partitioning hypergraphs using a new algorithmic version of the Lovász Local Lemma*

In his seminal result, Beck gave the first algorithmic version of the Lovász Local Lemma by giving polynomial time algorithms for 2‐coloring and partitioning uniform hypergraphs. His work was later generalized by Alon, and Molloy and Reed. Recently, Czumaj and Scheideler gave an efficient algorithm for 2‐coloring nonuniform hypergraphs. But the partitioning algorithm obtained based on their second paper only applies to a more limited range of hypergraphs, so much so that their work doesn't imply the result of Beck for the uniform case. Here we give an algorithmic version of the general form of the Local Lemma which captures (almost) all applications of the results of Beck and Czumaj and Scheideler, with an overall simpler proof. In particular, if H is a nonuniform hypergraph in which every edge e_i intersects at most |e_i|2^αk other edges of size at most k, for some small constant α, then we can find a partitioning of H in expected linear time. This result implies the result of Beck for uniform hypergraphs along with a speedup in his running time. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004     

The Smallest Parallelepiped of n Random Points and Peeling

Let n random points be given with uniform distribution in the d-dimensional unit cube [0,1]^d. The smallest parallelepiped A which includes all the n random points is dealt with. We investigate the asymptotic behavior of the volume of A as n tends to . Using a point process approach, we derive also the asymptotic behavior of the volumes of the k-th smallest parallelepipeds A _n ^(k) which are defined by iteration. Let A _n = A _n ⁽¹⁾ . Given A _n ^(k,-,1) delete the random points X _i which are on the boundary A _n ^(k,-,1) , and construct the smallest parallelepiped which includes the inner points of A _n ^(k,-,1) , this defines A _n ^(k) . This procedure is known as peeling of the parallelepiped A_n.     

Random walks on the torus with several generators

Given n vectors {_i} ∈ [0, 1)^d, consider a random walk on the d‐dimensional torus ??^d = ?^d/?^d generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*^k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*^k) ≥ C₁k^?n/2, where C₁ = C(n, d) is a constant. If the vectors are badly approximated by rationals (in a sense we will define), then D(Q*^k) ≤ C₂k^?n/2d for C₂ = C(n, d, _j) a constant. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Perfect fractional matchings in random hypergraphs

Given an r-uniform hypergraph H = (V, E) on |V| = n vertices, a real-valued function f:E→R⁺ is called a perfect fractional matching if Σ_vϵe f(e) ≤ 1 for all vϵV and Σ_eϵE f(e) = n/r. Considering a random r-uniform hypergraph process of n vertices, we show that with probability tending to 1 as n→ infinity, at the very moment t₀ when the last isolated vertex disappears, the hypergraph H_t0 has a perfect fractional matching. This result is clearly best possible. As a consequence, we derive that if p(n) = (ln n + w(n))/ , where w(n) is any function tending to infinity with n, then with probability tending to 1 a random r-uniform hypergraph on n vertices with edge probability p has a perfect fractional matching. Similar results hold also for random r-partite hypergraphs. © 1996 John Wiley & Sons, Inc.     

Two remarks on the Burr–Erdős conjecture

The Ramsey number r(H) of a graph H is the minimum positive integer N such that every two-coloring of the edges of the complete graph K_N on N vertices contains a monochromatic copy of H. A graph H is d-degenerate if every subgraph of H has minimum degree at most d. Burr and Erdős in 1975 conjectured that for each positive integer d there is a constant c_d such that r(H)≤c_dn for every d-degenerate graph H on n vertices. We show that for such graphs , improving on an earlier bound of Kostochka and Sudakov. We also study Ramsey numbers of random graphs, showing that for d fixed, almost surely the random graph G(n,d/n) has Ramsey number linear in n. For random bipartite graphs, our proof gives nearly tight bounds.     

The scaling window for a random graph with a given degree sequence

We consider a random graph on a given degree sequence D, satisfying certain conditions. Molloy and Reed defined a parameter Q = Q(D) and proved that Q = 0 is the threshold for the random graph to have a giant component. We introduce a new parameter R = R( \begin{align*}\mathcal {D}\end{align*}) and prove that if |Q| = O(n^‐1/3R^2/3) then, with high probability, the size of the largest component of the random graph will be of order Θ(n^2/3R^‐1/3). If |Q| is asymptotically larger than n^‐1/3R^2/3 then the size of the largest component is asymptotically smaller or larger than n^2/3R^‐1/3. Thus, we establish that the scaling window is |Q| = O(n^‐1/3R^2/3). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012