Random Graph Gn Research Articles

The expected relative error of the polyhedral approximation of the max-cut problem

We study the expected relative error of a linear relaxation of the max-cut problem in the random graph Gn,p. We prove that this error tends to 13 as n → ∞ of the edge probability p = p(n) is at least Ω(√logn/n), and tends to 1 if pn → ∞ and pn1−a → 0 for all a > 0.

Thresholds for classes of intersection graphs

For various previously studied classes of intersection graphs, the ranges of p = p(n) in which the random graph Gn,p almost surely falls in these classes are determined.

Complete Convergence of the Directed TSP

Consider the random directed graph Gn whose vertices X1, …, Xn are independent uniformly distributed over [0, 1]2. For 1 ≤ i < j ≤ n, the orientation of the edge XiXj is selected at random, independently for each edge and independently of the Xi's. Denote by Un the length of the shortest path through Gn. Then for some constant β > 0 and all ε > 0 we have ∑n≥1P(|n−1/2Un − β| > ε) < ∞.

Proportional graphs

AbstractLet S be a finite set of graphs and t a real number, 0 < t < 1. A (deterministic) graph G is (t, 5)‐proportional if for every H ∈ S, the number of induced subgraphs of G isomorphic to H equals the expected number of induced copies of H in the random graph Gn, t where n = |V(G)|. Let Sk = {all graphs on k vertices}, in particular S3 = {K3, P2, K2Kt, D3}. The notion of proportional graphs stems from the study of random graphs (Barbour, Karoński, and Ruciński, J Combinat. Th. Ser. B, 47, 125‐145, 1989; Janson and Nowicki, Prob. Th. Rel. Fields, to appear, Janson, Random Struct. Alg., 1, 15‐37, 1990) where it is shown that (t, S3)‐proportional graphs play a very special role; we thus call them simply t‐proportional. However, only a few ½‐proportional graphs on 8 vertices were known and it was an open problem whether there are any f‐proportional graphs with t ≠ ½ at all. In this paper, we show that there are infinitely many ½‐proportional graphs and that there are t‐proportional graphs with t≠. Both results are proved constructively. [We are not able to provide the latter construction for all f∈ Q∩(0,1), but the set of ts for which our construction works is dense in (0,1).] To support a conviction that the existence of (t, S3)‐proportional graphs was not quite obvious, we show that there are no (t, S4)‐proportional graphs.

Spanning maximal planar subgraphs of random graphs

AbstractWe study the threshold for the existence of a spanning maximal planar subgraph in the random graph Gn, p . We show that it is very near p = 1/n⅓ We also discuss the threshold for the existence of a spanning maximal outerplanar subgraph. This is very near p = 1/n½.

On the chromatic number of random graphs

AbstractWe consider random graphs Gn,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 logb n − 2 logb logb n + O(1)} where b = 1/(1 − p).

A random graph with a subcritical number of edges

A random graphGn(prob⁡(edge)=p)(p=c/n,0>c>1){G_n}(\operatorname {prob} (\operatorname {edge} ) = p)\;(p = c/n,\,0 > c > 1)onnnlabelled vertices is studied. There are obtained limiting distributions of the following characteristics: the lengths of the longest cycle and the longest path, the total size of unicyclic components, the number of cyclic vertices, the number of distinct component sizes, and the middle terms of the component-size order sequence. For instance, it is proved that, with probability approaching(1−c)1/2exp⁡(∑j=1lcj/2j){(1 - c)^{1/2}}\exp (\sum \nolimits _{j = 1}^l {{c^j}/2j)}asn→∞n \to \infty, the random graph does not have a cycle of length>l> l. Another result is that, with probability approaching11, the size of theν\nuth largest component either equals an integer closest toalog⁡(bn/νlog5/2n)a\;\log (bn/\nu \,{\log ^{5/2}}n),a=a(c)a = a(c),b=b(c)b = b(c), or is one less than this integer, provided thatν→∞\nu \to \inftyandν=o(n/log5/2n)\nu = o(n/{\log ^{5/2}}n).

Sharp concentration of the chromatic number on random graphsG n, p

The distribution of the chromatic number on random graphsG n, p is quite sharply concentrated. For fixedp it concentrates almost surely in √n ω(n) consecutive integers where ω(n) approaches infinity arbitrarily slowly. If the average degreepn is less thann 1/6, it concentrates almost surely in five consecutive integers. Large deviation estimates for martingales are used in the proof.

General deletion lemmas via the Harris inequality

We present a new approach to the deletion method that is based on the Harris inequality. We obtain deletion lemmas that are similar in spirit to those of Rodl and Rucinski, but hold for arbitrary decreasing properties. That is, we show that under appropriate conditions, with very high (‘Janson-like’) probability it suffices to delete a small fraction of the edges of a random graph Gn,p to ensure that a given decreasing property holds. We also obtain stronger results for decreasing properties that only depend on edges involved in copies of some given graph H. As an application of our methods, we present a new deletion lemma that concerns local subgraph counts, i.e., the number of copies of H each edge or vertex of Gn,p is contained in.