Random Graph Gn Research Articles

Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors

We study how the spectral gap of the normalized Laplacian of a random graph changes when an edge is added to or removed from the graph. There are known examples of graphs where, perhaps counter-intuitively, adding an edge can decrease the spectral gap, a phenomenon that is analogous to Braess's paradox in traffic networks. We show that this is often the case in random graphs in a strong sense. More precisely, we show that for typical instances of Erdi¾?s-Renyi random graphs Gn, p with constant edge density p∈0,1, the addition of a random edge will decrease the spectral gap with positive probability, strictly bounded away from zero. To do this, we prove a new delocalization result for eigenvectors of the Laplacian of Gn, p, which might be of independent interest. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 584-611, 2017

Saturation in random graphs

A graph H is Ks-saturated if it is a maximal Ks-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 Ks-saturated graph was determined over 50 years ago by Zykov and independently by Erdi¾?s, Hajnal and Moon. In this paper, we study the random analog of this problem: minimizing the number of edges in a maximal Ks-free subgraph of the Erdi¾?s-Renyi random graph Gn, 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

Graph Decomposition and Parity

Motivated by a recent extension of the zero-one law by Kolaitis and Kopparty, we study the distribution of the number of copies of a fixed disconnected graph in the random graph Gn,p. We use an idea of graph decompositions to give a sufficient condition for this distribution to tend to uniform modulo q. We determine the asymptotic distribution of all fixed two-component graphs in Gn,p for all q, and we give infinite families of many-component graphs with a uniform asymptotic distribution for all q. We also prove a negative result that no recursive proof of the simplest form exists for a uniform asymptotic distribution for arbitrary graphs.

The matching energy of random graphs

The matching energy of a graph was introduced by Gutman and Wagner, which is defined as the sum of the absolute values of the roots of the matching polynomial of the graph. For the random graph Gn,p of order n with fixed probability p∈(0,1), Gutman and Wagner (2012) proposed a conjecture that the expectation of the matching energy of Gn,p is asymptotically equal to 8p3πn32. In this paper, using analytical tools, we confirm this conjecture by obtaining a stronger result that the matching energy of Gn,p is asymptotically almost surely equal to 8p3πn32.

A note on the existence of fractional f-factors in random graphs

Let G = G n,p be a binomial random graph with n vertices and edge probability p = p(n), and f be a nonnegative integer-valued function defined on V(G) such that $$d_G^h (x) = \sum\limits_{x \in e} {h(e)}$$ for every x ∈ V(G). An fractional f-indicator function is an function h that assigns to each edge of a graph G a number h(e) in [0, 1] so that for each vertex x, we have d (x) = f(x), where 0 < a ≤ f(x) ≤ b < np − 2√nplogn is the fractional degree of x in G. Set E h = {e: e ∈ E(G) and h(e) ≠ 0}. If G h is a spanning subgraph of G such that E(G h ) = E h , then G h is called an fractional f-factor of G. In this paper, we prove that for any binomial random graph G n,p with $$p \geqslant n^{ - \tfrac{2} {3}}$$ , almost surely G n,p contains an fractional f-factor.

On the Spread of Random Graphs

The spread of a connected graph G was introduced by Alon, Boppana and Spencer [1], and measures how tightly connected the graph is. It is defined as the maximum over all Lipschitz functions f on V(G) of the variance of f(X) when X is uniformly distributed on V(G). We investigate the spread for certain models of sparse random graph, in particular for random regular graphs G(n,d), for Erdős–Rényi random graphs Gn,p in the supercritical range p&gt;1/n, and for a ‘small world’ model. For supercritical Gn,p, we show that if p=c/n with c&gt;1 fixed, then with high probability the spread of the giant component is bounded, and we prove corresponding statements for other models of random graphs, including a model with random edge lengths. We also give lower bounds on the spread for the barely supercritical case when p=(1+o(1))/n. Further, we show that for d large, with high probability the spread of G(n,d) becomes arbitrarily close to that of the complete graph $\mathsf{K}_n$.

Sandwiching a densest subgraph by consecutive cores

In this paper, we show that in the random graph Gn,c/n, with high probability, there exists an integer ki¾? such that a subgraph of Gn,c/n, whose vertex set differs from a densest subgraph of Gn,c/n by Olog2n vertices, is sandwiched by the ki¾? and the ki¾?+1-core, for almost all sufficiently large c. We determine the value of ki¾?. We also prove that a, the threshold of the k-core being balanced coincides with the threshold that the average degree of the k-core is at most 2k-1, for all sufficiently large k; b with high probability, there is a subgraph of Gn,c/n whose density is significantly denser than any of its nonempty cores, for almost all sufficiently large c > 0. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 341-360, 2015

POISSON APPROXIMATION FOR THE NUMBER OF OPIES OF A FIXED GRAPH IN A RANDOM d-REGULAR GRAPH

Let W be the number of copies of a fixed graph H in a random dregular graph Gn,d. In this paper, we give the bound on Poisson approximation of W by using the Stein-Chen method.

An approximate blow-up lemma for sparse pseudorandom graphs

We state a sparse approximate version of the blow-up lemma, showing that regular partitions in sufficiently pseudorandom graphs behave almost like complete partite graphs for embedding graphs with maximum degree Δ. We show that (p,γ)-jumbled graphs, with γ=o(pmax(2Δ,Δ+3/2)n), are “sufficiently pseudorandom”.The approach extends to random graphs Gn,p with p≫(lognn)1/Δ.

Largest sparse subgraphs of random graphs

For the Erdős–Rényi random graph Gn,p, we give a precise asymptotic formula for the size αˆt(Gn,p) of a largest vertex subset in Gn,p that induces a subgraph with average degree at most t, provided that p=p(n) is not too small and t=t(n) is not too large. In the case of fixed t and p, we find that this value is asymptotically almost surely concentrated on at most two explicitly given points. This generalises a result on the independence number of random graphs. For both the upper and lower bounds, we rely on large deviations inequalities for the binomial distribution.

Average-case complexity of backtrack search for coloring sparse random graphs

We investigate asymptotically the expected number of steps taken by backtrack search for k-coloring random graphs Gn,p(n) or proving non-k-colorability, where p(n) is an arbitrary sequence tending to 0, and k is constant. Contrary to the case of constant p, where the expected runtime is known to be O(1), we prove that here the expected runtime tends to infinity. We establish how the asymptotic behavior of the expected number of steps depends on the sequence p(n). In particular, for p(n)=d/n, where d is a constant, the runtime is always exponential, but it can be also polynomial if p(n) decreases sufficiently slowly, e.g. for p(n)=1/lnn.

Rainbow hamilton cycles in random graphs

AbstractOne of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdős‐Rényi random graph Gn,p is around . Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3‐uniform hypergraph by connecting 3‐uniform hypergraphs to edge‐colored graphs.In this work, we consider that setting of edge‐colored graphs, and prove a result which achieves the best possible first order constant. Specifically, when the edges of Gn,p are randomly colored from a set of (1 + o(1))n colors, with , we show that one can almost always find a Hamilton cycle which has the additional property that all edges are distinctly colored (rainbow).Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 44, 328‐354, 2014

Coloring random graphs online without creating monochromatic subgraphs

AbstractConsider the following random process: The vertices of a binomial random graph Gn,p are revealed one by one, and at each step only the edges induced by the already revealed vertices are visible. Our goal is to assign to each vertex one from a fixed number r of available colors immediately and irrevocably without creating a monochromatic copy of some fixed graph F in the process. Our first main result is that for any F and r, the threshold function for this problem is given by p0(F,r,n) = n‐1/m*1(F,r), where m*1(F,r) denotes the so‐called online vertex‐Ramsey density of F and r. This parameter is defined via a purely deterministic two‐player game, in which the random process is replaced by an adversary that is subject to certain restrictions inherited from the random setting. Our second main result states that for any F and r, the online vertex‐Ramsey density m*1(F,r) is a computable rational number.Our lower bound proof is algorithmic, i.e., we obtain polynomial‐time online algorithms that succeed in coloring Gn,p as desired with probability 1 ‐ o(1) for any p(n) = o(n‐1/m*1(F,r)). © 2012 Wiley Periodicals, Inc. Random Struct. Alg. 44, 419–464, 2014

Extremal subgraphs of random graphs

AbstractWe prove that there is a constant c &gt; 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 Gn,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 ≤ $(\matrix{ n \cr 2 \cr } )$ /2, is “nearly unique”. More precisely, given a maximum cut C of Gn,M, we can obtain all maximum cuts by moving at most \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{O}(\sqrt{n^3/M})\end{align*}\end{document} vertices between the parts of C. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

Sparse random graphs: Eigenvalues and eigenvectors

AbstractIn this paper, we prove the semi‐circular law for the eigenvalues of regular random graph Gn,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 Erdős–Rényi random graph G(n,p), answering a question raised by Dekel–Lee–Linial. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

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

AbstractWe 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 Gn,p (above the connectivity threshold) and for random regular graphs Gr,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 Gn,p and Gr we give the value of t*, and the size of Γ(t). For Gr, 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 Dn,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

Largest sparse subgraphs of random graphs

Abstract For the Erdős-Renyi random graph Gn,p, we consider the order of a largest vertex subset that induces a subgraph with average degree at most t. For the case when both p and t are fixed, this value is asymptotically almost surely concentrated on at most two explicitly given points. This generalises a result on the independence number of random graphs. For both the upper and lower bounds, we rely on large deviations inequalities for the binomial distribution.

First Passage Percolation on the Erdős–Rényi Random Graph

In this paper we explore first passage percolation (FPP) on the Erdős–Rényi random graph Gn(pn), where we assign independent random weights, having an exponential distribution with rate 1, to the edges. In the sparse regime, i.e., when npn → λ &gt; 1, we find refined asymptotics both for the minimal weight of the path between uniformly chosen vertices in the giant component, as well as for the hopcount (i.e., the number of edges) on this minimal weight path. More precisely, we prove a central limit theorem for the hopcount, with asymptotic mean and variance both equal to (λ log n)/(λ − 1). Furthermore, we prove that the minimal weight centred by (log n)/(λ − 1) converges in distribution.We also investigate the dense regime, where npn → ∞. We find that although the base graph is ultra-small (meaning that graph distances between uniformly chosen vertices are o(log n)), attaching random edge weights changes the geometry of the network completely. Indeed, the hopcount Hn satisfies the universality property that whatever the value of pn, Hn/log n → 1 in probability and, more precisely, (Hn − βn log n)/, where βn = λn/(λn − 1), has a limiting standard normal distribution. The constant βn can be replaced by 1 precisely when λn ≫ , a case that has appeared in the literature (under stronger conditions on λn) in [4, 13]. We also find lower bounds for the maximum, over all pairs of vertices, of the optimal weight and hopcount.This paper continues the investigation of FPP initiated in [4] and [5]. Compared to the setting on the configuration model studied in [5], the proofs presented here are much simpler due to a direct relation between FPP on the Erdős–Rényi random graph and thinned continuous-time branching processes.

Approximate Hamilton decompositions of random graphs

We show that if pn ≫ log n the binomial random graph Gn,p has an approximate Hamilton decomposition. More precisely, we show that in this range Gn,p contains a set of edge-disjoint Hamilton cycles covering almost all of its edges. This is best possible in the sense that the condition that pn ≫ log n is necessary. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012 © 2012 Wiley Periodicals, Inc.

The independent domination number of a random graph

We prove a two-point concentration for the independent domination number of the random graph Gn,p provided p 2 ln(n) � 64ln((lnn)=p). occurs asymptotically almost surely (a.a.s.) if P(Gn;p has property A) ! 1 as n ! 1 . See Bollobas (2) for notation and terminology. Weber (7) showed if p = 1=2 then a.a.s. (Gn;p) is either blog2 n − log2(log2 nlnn)c + 1 or blog2 n − log2(log2 nlnn)c + 2 and a.a.s. i(Gn;p)