On MAXCUT in strictly supercritical random graphs,and coloring of random graphs and random tournaments

We use a theorem by Ding, Lubetzky, and Peres describing the structure of the giant component of random graphs in the strictly supercritical regime, in order to determine the typical size of MAXCUT of in terms of ɛ. We then apply this result to prove the following conjecture by Frieze and Pegden. For every , there exists such that w.h.p. is not homomorphic to the cycle on vertices. We also consider the coloring properties of biased random tournaments. A p‐random tournament on n vertices is obtained from the transitive tournament by reversing each edge independently with probability p. We show that for the chromatic number of a p‐random tournament behaves similarly to that of a random graph with the same edge probability. To treat the case we use the aforementioned result on MAXCUT and show that in fact w.h.p. one needs to reverse edges to make it 2‐colorable.     

The cover time of random geometric graphs

We study the cover time of random geometric graphs. Let $I(d)=[0,1]^{d}$ denote the unit torus in d dimensions. Let $D(x,r)$ denote the ball (disc) of radius r. Let $\Upsilon_d$ be the volume of the unit ball $D(0,1)$ in d dimensions. A random geometric graph $G=G(d,r,n)$ in d dimensions is defined as follows: Sample n points V independently and uniformly at random from $I(d)$ . For each point x draw a ball $D(x,r)$ of radius r about x. The vertex set $V(G)=V$ and the edge set $E(G)=\{\{v,w\}: w\ne v,\,w\in D(v,r)\}$ . Let $G(d,r,n),\,d\geq 3$ be a random geometric graph. Let $C_G$ denote the cover time of a simple random walk on G. Let $c>1$ be constant, and let $r=(c\log n/(\Upsilon_dn))^{1/d}$ . Then whp the cover time satisfies © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 324–349, 2011     

随机偏好连接图的中心极限定理

我们研究了一类具有随机顶点和边的随机连接图模型, 其中顶点的随机性由一个Poisson 点过程所决定, 边的随机性由一个概率连接函数所决定. 我们得到了带偏好的随机连接图模型的关于所有随机边的长度和的一个中心极限定理.     

Extremal subgraphs of random graphs

We prove that there is a constant c > 0, such that whenever p ≥ n^‐c, with probability tending to 1 when n goes to infinity, every maximum triangle‐free subgraph of the random graph G_n,p is bipartite. This answers a question of Babai, Simonovits and Spencer (Babai et al., J Graph Theory 14 (1990) 599–622). The proof is based on a tool of independent interest: we show, for instance, that the maximum cut of almost all graphs with M edges, where M ? n and M ≤ /2, is “nearly unique”. More precisely, given a maximum cut C of G_n,M, we can obtain all maximum cuts by moving at most \begin{align*}\mathcal{O}(\sqrt{n^3/M})\end{align*} vertices between the parts of C. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

The diameter of sparse random graphs

We derive an expression of the form c ln n + o(ln n) for the diameter of a sparse random graph with a specified degree sequence. The result holds asymptotically almost surely, assuming that certain convergence and supercriticality conditions are met, and is applicable to the classical random graph G_n,p with np = Θ(1) + 1, as well as certain random power law graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Quasi‐random graphs with given degree sequences

It is now known that many properties of the objects in certain combinatorial structures are equivalent, in the sense that any object possessing any of the properties must of necessity possess them all. These properties, termed quasirandom, have been described for a variety of structures such as graphs, hypergraphs, tournaments, Boolean functions, and subsets of Z _n, and most recently, sparse graphs. In this article, we extend these ideas to the more complex case of graphs which have a given degree sequence. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 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.     

Sparse pseudo‐random graphs are Hamiltonian

In this article we study Hamilton cycles in sparse pseudo‐random graphs. We prove that if the second largest absolute value λ of an eigenvalue of a d‐regular graph G on n vertices satisfies and n is large enough, then G is Hamiltonian. We also show how our main result can be used to prove that for every c >0 and large enough n a Cayley graph X (G,S), formed by choosing a set S of c log⁵ n random generators in a group G of order n, is almost surely Hamiltonian. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 17–33, 2003     

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     

Saturation in random graphs

A graph H is K_s ‐saturated if it is a maximal K_s ‐free graph, i.e., H contains no clique on s vertices, but the addition of any missing edge creates one. The minimum number of edges in a K_s ‐saturated graph was determined over 50 years ago by Zykov and independently by Erd?s, Hajnal and Moon. In this paper, we study the random analog of this problem: minimizing the number of edges in a maximal K_s ‐free subgraph of the Erd?s‐Rényi random graph G (n, p ). We give asymptotically tight estimates on this minimum, and also provide exact bounds for the related notion of weak saturation in random graphs. Our results reveal some surprising behavior of these parameters. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 169–181, 2017     

Pancyclic subgraphs of random graphs

An n‐vertex graph is called pancyclic if it contains a cycle of length t for all 3≤t≤n. In this article, we study pancyclicity of random graphs in the context of resilience, and prove that if p>n^?1/2, then the random graph G(n, p) a.a.s. satisfies the following property: Every Hamiltonian subgraph of G(n, p) with more than edges is pancyclic. This result is best possible in two ways. First, the range of p is asymptotically tight; second, the proportion of edges cannot be reduced. Our theorem extends a classical theorem of Bondy, and is closely related to a recent work of Krivelevich et al. The proof uses a recent result of Schacht (also independently obtained by Conlon and Gowers). © 2011 Wiley Periodicals, Inc.     

The chromatic number of dense random graphs

The chromatic number of a graph G is defined as the minimum number of colors required for a vertex coloring where no two adjacent vertices are colored the same. The chromatic number of the dense random graph where is constant has been intensively studied since the 1970s, and a landmark result by Bollobás in 1987 first established the asymptotic value of . Despite several improvements of this result, the exact value of remains open. In this paper, new upper and lower bounds for are established. These bounds are the first ones that match each other up to a term of size o(1) in the denominator: they narrow down the coloring rate of to an explicit interval of length o(1), answering a question of Kang and McDiarmid.     

Sparse graphs: Metrics and random models

Recently, Bollobás, Janson and Riordan introduced a family of random graph models producing inhomogeneous graphs with n vertices and Θ(n) edges whose distribution is characterized by a kernel, i.e., a symmetric measurable function κ: [0, 1]² → [0, ∞). To understand these models, we should like to know when different kernels κ give rise to “similar” graphs, and, given a real‐world network, how “similar” is it to a typical graph G(n, κ) derived from a given kernel κ. The analogous questions for dense graphs, with Θ(n²) edges, are answered by recent results of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi, who showed that several natural metrics on graphs are equivalent, and moreover that any sequence of graphs converges in each metric to a graphon, i.e., a kernel taking values in [0, 1]. Possible generalizations of these results to graphs with o(n²) but ω(n) edges are discussed in a companion article [Bollobás and Riordan, London Math Soc Lecture Note Series 365 (2009), 211–287]; here we focus only on graphs with Θ(n) edges, which turn out to be much harder to handle. Many new phenomena occur, and there are a host of plausible metrics to consider; many of these metrics suggest new random graph models and vice versa. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 1‐38, 2011     

On the chromatic number of random graphs

We consider random graphs G_n,p with fixed edge-probability p. We refine an argument of Bollobás to show that almost all such graphs have chromatic number equal to n/{2 log_b n ? 2 log_b log_b n + O(1)} where b = 1/(1 ? p).     

Random dense bipartite graphs and directed graphs with specified degrees

Let s and t be vectors of positive integers with the same sum. We study the uniform distribution on the space of simple bipartite graphs with degree sequence s in one part and t in the other; equivalently, binary matrices with row sums s and column sums t . In particular, we find precise formulae for the probabilities that a given bipartite graph is edge‐disjoint from, a subgraph of, or an induced subgraph of a random graph in the class. We also give similar formulae for the uniform distribution on the set of simple directed graphs with out‐degrees s and in‐degrees t . In each case, the graphs or digraphs are required to be sufficiently dense, with the degrees varying within certain limits, and the subgraphs are required to be sufficiently sparse. Previous results were restricted to spaces of sparse graphs. Our theorems are based on an enumeration of bipartite graphs avoiding a given set of edges, proved by multidimensional complex integration. As a sample application, we determine the expected permanent of a random binary matrix with row sums s and column sums t . © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Sparse random graphs: Eigenvalues and eigenvectors

In this paper, we prove the semi‐circular law for the eigenvalues of regular random graph G_n,d in the case d →∞, complementing a previous result of McKay for fixed d. We also obtain a upper bound on the infinity norm of eigenvectors of Erd?s–Rényi random graph G(n,p), answering a question raised by Dekel–Lee–Linial. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Critical window for the vacant set left by random walk on random regular graphs

We consider the simple random walk on a random d ‐regular graph with n vertices, and investigate percolative properties of the set of vertices not visited by the walk until time \begin{align*}\left\lfloor un \right\rfloor\end{align*}, where u > 0 is a fixed positive parameter. It was shown in ?erný et al., (Ann Inst Henri Poincaré Probab Stat 47 (2011) 929–968) that this so‐called vacant set exhibits a phase transition at u = u_?: there is a giant component if u < u_? and only small components when u > u_?. In this paper we show the existence of a critical window of size n^‐1/3 around u_?. In this window the size of the largest cluster is of order n^2/3. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013