Binomial Random Graph Research Articles

On Pósa's Conjecture for Random Graphs

The famous Posa conjecture states that every graph of minimum degree at least $2n/3$ contains the square of a Hamilton cycle. This has been proved for large $n$ by Komlos, Sarkozy, and Szemeredi. Here we prove that if $p \ge n^{-1/2+\varepsilon}$, then asymptotically almost surely, the binomial random graph $G_{n,p}$ contains the square of a Hamilton cycle. This provides an “approximate threshold” for the property in the sense that the result fails to hold if $p\le n^{-1/2}$.

On the path-avoidance vertex-coloring game

Abstract For any graph F and any integer r ⩾ 2 , the online vertex-Ramsey density of F and r, denoted m ⁎ ( F , r ) , is a parameter defined via a deterministic two-player Ramsey-type game (Painter vs. Builder). This parameter was introduced in a recent paper [Mutze, T., T. Rast and R. Spohel, Coloring random graphs online without creating monochromatic subgraphs, in: Proceedings of the 22nd annual ACM-SIAM Symposium on Discrete Algorithms (SODA ʼ11), 2011, pp. 145–158], where it was shown that the online vertex-Ramsey density determines the threshold of a similar probabilistic one-player game (Painter vs. the binomial random graph G n , p ). For a large class of graphs F, including cliques, cycles, complete bipartite graphs, hypercubes, wheels, and stars of arbitrary size, a simple greedy strategy is optimal for Painter and closed formulas for m ⁎ ( F , r ) are known. In this work we show that in the innocent-looking case where F = P l is a (long) path, the picture is very different. As it turns out, for this family of graphs the greedy strategy fails quite badly, and the parameter m ⁎ ( F , r ) exhibits a much more complex behavior than one might expect in view of the other examples mentioned.

On the Path-Avoidance Vertex-Coloring Game

For any graph $F$ and any integer $r\geq 2$, the online vertex-Ramsey density of $F$ and $r$, denoted $m^*(F,r)$, is a parameter defined via a deterministic two-player Ramsey-type game (Painter vs. Builder). This parameter was introduced in a recent paper [arXiv:1103.5849], where it was shown that the online vertex-Ramsey density determines the threshold of a similar probabilistic one-player game (Painter vs. the binomial random graph $G_{n,p}$). For a large class of graphs $F$, including cliques, cycles, complete bipartite graphs, hypercubes, wheels, and stars of arbitrary size, a simple greedy strategy is optimal for Painter and closed formulas for $m^*(F,r)$ are known. In this work we show that for the case where $F=P_\ell$ is a long path, the picture is very different. It is not hard to see that $m^*(P_\ell,r)= 1-1/k^*(P_\ell,r)$ for an appropriately defined integer $k^*(P_\ell,r)$, and that the greedy strategy gives a lower bound of $k^*(P_\ell,r)\geq \ell^r$. We construct and analyze Painter strategies that improve on this greedy lower bound by a factor polynomial in $\ell$, and we show that no superpolynomial improvement is possible.

On an anti-Ramsey property of random graphs

Abstract For graphs G and H, let G → p m c H denote the property that, for every proper edge-colouring of G, with an arbitrary number of colours, there is a multicoloured, or rainbow, copy of H in G, that is a copy of H with no two edges of the same colour. We consider the problem of establishing the threshold p H m c = p H m c ( n ) of this property for the binomial random graph G ( n , p ) .

A concentration result with application to subgraph count

AbstractLet H = (V,E) be a k ‐uniform hypergraph with a vertex set V and an edge set E. Let V p be constructed by taking every vertex in V independently with probability p. Let X be the number of edges in E that are contained in V p. We give a condition that guarantees the concentration of X within a small interval around its mean. The applicability of this result is demonstrated by deriving new sub‐Gaussian tail bounds for the number of copies of small complete and complete bipartite graphs in the binomial random graph. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012

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.

Regular induced subgraphs of a random Graph

AbstractAn old problem of Erdős, Fajtlowicz, and Staton asks for the order of a largest induced regular subgraph that can be found in every graph on $n$ vertices. Motivated by this problem, we consider the order of such a subgraph in a typical graph on $n$ vertices, i.e., in a binomial random graph $G(n,1/2)$. We prove that with high probability a largest induced regular subgraph of $G(n,1/2)$ has about $n^{2/3}$ vertices. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 235–250, 2011

The natural connectivity of colored random graphs

The natural connectivity as a robustness measure of complex network has been proposed recently. It can be regarded as the average eigenvalue obtained from the graph spectrum. In this paper, we introduce an inhomogeneous random graph model, G(n, {ci}, {pi}), and investigate its natural connectivity. Binomial random graph ... . Simulations are performed to validate our theoretical results.

Local Resilience and Hamiltonicity Maker–Breaker Games in Random Regular Graphs

For an increasing monotone graph propertythelocal resilienceof a graphGwith respect tois the minimalrfor which there exists a subgraphH⊆Gwith all degrees at mostr, such that the removal of the edges ofHfromGcreates a graph that does not possess. This notion, which was implicitly studied for somead hocproperties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the binomial random graph model(n, p) and some families of pseudo-random graphs with respect to several graph properties, such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudo-randomregulargraphs of constant degree. We investigate the local resilience of the typical randomd-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular, we prove that for every positive ϵ and large enough values ofd, with high probability, the local resilience of the randomd-regular graph,n, d, with respect to being Hamiltonian, is at least (1−ϵ)d/6. We also prove that for the binomial random graph model(n, p), for every positive ϵ > 0 and large enough values ofK, ifp>$\frac{K\ln n}{n}$then, with high probability, the local resilience of(n, p) with respect to being Hamiltonian is at least (1−ϵ)np/6. Finally, we apply similar techniques to positional games, and prove that ifdis large enough then, with high probability, a typical randomd-regular graphGis such that, in the unbiased Maker–Breaker game played on the edges ofG, Maker has a winning strategy to create a Hamilton cycle.

The logic of random regular graphs

The first order language of graphs is a formal language in which one can express many properties of graphs — known as first order properties. The classic Zero-One law for random graphs states that if p is some constant probability then for every first order property the limiting probability of the binomial random graph G(n, p) having this property is either zero or one. The case of sparse random graphs has also been studied in detail for the binomial random graph model. We obtain results for random regular graphs that match the main results for G(n, p). In particular we prove that if the degree d is linear in the number of vertices n, or if d = n for 0 < α < 1 irrational, then the random d-regular graph Gn,d obeys the Zero-One law. On the contrary, if d = n for rational 0 < α < 1, then there is a (theoretically explicit) first order property with no limiting probability in Gn,d.

Investigating the Properties of a Social Bookmarking and Tagging Network

Social networks and collaborative tagging systems are rapidly gaining popularity as a primary means for storing and sharing data among friends, family, colleagues, or perfect strangers as long as they have common interests. del.icio.us3 is a social network where people store and share their personal bookmarks. Most importantly, users tag their bookmarks for ease of information dissemination and later look up. However, it is the friendship links that make del.icio.us a social network. They exist independently of the set of bookmarks that belong to the users and have no relation to the tags typically assigned to the bookmarks. To study the interaction among users, the strength of the existing links and their hidden meaning, we introduce implicit links in the network. These links connect only highly “similar” users. Here, similarity can reflect different aspects of the user’s profile that makes her similar to any other user, such as number of shared bookmarks, or similarity of their tags clouds. The authors investigate the question whether friends have common interests, they gain additional insights on the strategies that users use to assign tags to their bookmarks, and they demonstrate that the graphs formed by implicit links have unique properties differing from binomial random graphs or random graphs with an expected power-law degree distribution.

Random regular graphs of non-constant degree: Concentration of the chromatic number

In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model G n , d for d = o ( n 1 / 5 ) is concentrated in two consecutive values, thus extending a previous result of Achlioptas and Moore. This concentration phenomena is very similar to that of the binomial random graph model G ( n , p ) with p = d n . Our proof is largely based on ideas of Alon and Krivelevich who proved this two-point concentration result for G ( n , p ) for p = n − δ where δ > 1 / 2 . The main tool used to derive such a result is a careful analysis of the distribution of edges in G n , d , relying both on the switching technique and on bounding the probability of exponentially small events in the configuration model.

Gradient networks

Gradient networks are defined (Toroczkai and Bassler 2004 Nature 428 716) as directed graphs formed by local gradients of a scalar field distributed on the nodes of a substrate network G. We present the derivation for some of the general properties of gradient graphs and give an exact expression for the in-degree distribution of the gradient network when the substrate is a binomial (Erds–Rényi) random graph, , and the scalars are independent identically distributed (i.i.d.) random variables. We show that in the limit for , i.e., gradient networks become scale-free graphs up to a cut-off degree. This paper presents the detailed derivation of the results announced in Toroczkai and Bassler (2004 Nature 428 716).

Turán's theorem for pseudo-random graphs

The generalized Turán number ex ( G , H ) of two graphs G and H is the maximum number of edges in a subgraph of G not containing H. When G is the complete graph K m on m vertices, the value of ex ( K m , H ) is ( 1 − 1 / ( χ ( H ) − 1 ) + o ( 1 ) ) ( m 2 ) , where o ( 1 ) → 0 as m → ∞ , by the Erdős–Stone–Simonovits theorem. In this paper we give an analogous result for triangle-free graphs H and pseudo-random graphs G. Our concept of pseudo-randomness is inspired by the jumbled graphs introduced by Thomason [A. Thomason, Pseudorandom graphs, in: Random Graphs '85, Poznań, 1985, North-Holland, Amsterdam, 1987, pp. 307–331. MR 89d:05158]. A graph G is ( q , β ) -bi-jumbled if | e G ( X , Y ) − q | X | | Y | | ⩽ β | X | | Y | for every two sets of vertices X , Y ⊂ V ( G ) . Here e G ( X , Y ) is the number of pairs ( x , y ) such that x ∈ X , y ∈ Y , and x y ∈ E ( G ) . This condition guarantees that G and the binomial random graph with edge probability q share a number of properties. Our results imply that, for example, for any triangle-free graph H with maximum degree Δ and for any δ > 0 there exists γ > 0 so that the following holds: any large enough m-vertex, ( q , γ q Δ + 1 / 2 m ) -bi-jumbled graph G satisfies ex ( G , H ) ⩽ ( 1 − 1 χ ( H ) − 1 + δ ) | E ( G ) | .

The Turn Theorem for Random Graphs

The aim of this paper is to prove a Turan-type theorem for random graphs. For $\gamma >0$ and graphs $G$ and $H$, write $G\to_\gamma H$ if any $\gamma$-proportion of the edges of $G$ spans at least one copy of $H$ in $G$. We show that for every graph $H$ and every fixed real $\delta>0$, almost every graph $G$ in the binomial random graph model $\cG(n,q)$, with $q=q(n)\gg((\log n)^4/n)^{1/d(H)}$, satisfies $G\to_{(\chi(H)-2)/(\chi(H)-1)+\delta}H$, where as usual $\chi(H)$ denotes the chromatic number of $H$ and $d(H)$ is the ‘degeneracy number’ of $H$.Since $K_l$, the complete graph on $l$ vertices, is $l$-chromatic and $(l-1)$-degenerate, we infer that for every $l\geq2$ and every fixed real $\delta>0$, almost every graph $G$ in the binomial random graph model $\cG(n,q)$, with $q=q(n)\gg((\log n)^4/n)^{1/(l-1)}$, satisfies $G\to_{(l-2)/(l-1)+\delta}K_l$.

Solitary subgraphs of random graphs

Let G be a graph and let G(n,p) be the binomial random graph with n vertices and edge probability p. We consider copies of G in G(n,p) , vertex disjoint from all other such copies. For a strictly balanced graph G, initially, every copy of G in G(n,p) is solitary. Suen (Random Struct. Algorithms 1 (1990) 231–242) established a second (disappearance) threshold for a subclass of strictly balanced graphs. In this paper we extend his result to a more general case.

Threshold functions for asymmetric Ramsey properties involving cycles

We consider the binomial random graph Gp and determine a sharp threshold function for the edge-Ramsey propertyfor all l1,…,lr, where Cl denotes the cycle of length l. As deterministic consequences of our results, we prove the existence of sparse graphs having the above Ramsey property as well as the existence of infinitely many critical graphs with respect to the property above. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11, 245–276, 1997

Threshold functions for asymmetric Ramsey properties involving cycles

We consider the binomial random graph Gp and determine a sharp threshold function for the edge-Ramsey propertyfor all l1,…,lr, where Cl denotes the cycle of length l. As deterministic consequences of our results, we prove the existence of sparse graphs having the above Ramsey property as well as the existence of infinitely many critical graphs with respect to the property above. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11, 245–276, 1997

A note on local colorings of graphs

Recently, Rödl and Ruciński [5,6] proved the following threshold result about Ramsey properties of random graphs. Let K( n, p) be the binomial random graph obtained from the complete graph K( n) by independent deletion of each edge with probability 1 - p. We write F → ( G) r if for every r-coloring of the edges of F there is a monochromatic copy of G.

Random graphs with monochromatic triangles in every edge coloring

AbstractWe prove that for every r ⩾ 2 there is C &gt; 0 such that if p Cn−1/2 then almost surely every r‐coloring of the edges of the binomial random graph K(n, p) results in a monochromatic triangle. © 1994 John Wiley &amp; Sons, Inc.