Independence numbers of random subgraphs of a distance graph

We consider the so-called distance graph G(n, 3, 1), whose vertices can be identified with three-element subsets of the set {1, 2,..., n}, two vertices being joined by an edge if and only if the corresponding subsets have exactly one common element. We study some properties of random subgraphs of G(n, 3, 1) in the Erd?s–Rényi model, in which each edge is included in the subgraph with some given probability p independently of the other edges. We find the asymptotics of the independence number of a random subgraph of G(n, 3, 1).     

Spanning subgraphs of random graphs

We propose a problem concerning the determination of the threshold function for the edge probability that guarantees, almost surely, the existence of various sparse spanning subgraphs in random graphs. We prove some bounds and demonstrate them in the cases of ad-cube and a two dimensional lattice.     

Extremal subgraphs of random graphs

We shall prove that if L is a 3-chromatic (so called “forbidden”) graph, and —Rⁿ is a random graph on n vertices, whose edges are chosen independently, with probability p, and —Bⁿ is a bipartite subgraph of Rⁿ of maximum size, —Fⁿ is an L-free subgraph of Rⁿ of maximum size, then (in some sense) Fⁿ and Bⁿ are very near to each other: almost surely they have almost the same number of edges, and one can delete O_p(1) edges from Fⁿ to obtain a bipartite graph. Moreover, with p = 1/2 and L any odd cycle, Fⁿ is almost surely bipartite.     

Small subgraphs of random regular graphs

The main aim of this short paper is to answer the following question. Given a fixed graph H, for which values of the degree d does a random d-regular graph on n vertices contain a copy of H with probability close to one?     

Independence numbers of product graphs

This paper deals with the problem of determining the independence number for the strong graph-product, especially for odd cycles. Using the concepts of a point-symmetric graph and the clique-number and introducing the notion of an independence graph, we extend and generalize some results of Hales [3] for cycle-products of power three.     

Spanning maximal planar subgraphs of random graphs

We study the threshold for the existence of a spanning maximal planar subgraph in the random graph G_{n, p} . We show that it is very near p = 1/n^? We also discuss the threshold for the existence of a spanning maximal outerplanar subgraph. This is very near p = 1/n^½.     

OnK 4-free subgraphs of random graphs

For 0<1 and graphsG andH, writeGH if any -proportion of the edges ofG spans at least one copy ofH inG. As customary, writeK ^r for the complete graph onr vertices. We show that for every fixed real >0 there exists a constantC=C() such that almost every random graphG _n,p withp=p(n)Cn ^–2/5 satisfiesG _{n,p 2/3+} K ⁴. The proof makes use of a variant of Szemerédi's regularity lemma for sparse graphs and is based on a certain superexponential estimate for the number of pseudo-random tripartite graphs whose triangles are not too well distributed. Related results and a general conjecture concerningH-free subgraphs of random graphs in the spirit of the Erds-Stone theorem are discussed.The first author was partially supported by FAPESP (Proc. 93/0603-1) and by CNPq (Proc. 300334/93-1 and ProTeM-CC-II Project ProComb). Part of this work was done while the second author was visiting the University of São Paulo, supported by FAPESP (Proc. 94/4276-8). The third author was partially supported by the NSF grant DMS-9401559.     

Matchings in random spanning subgraphs of cubelike graphs

A question about the evolution of random spanning subgraphs G _p of bipartite regular so called cubelike graphs G is considered. It is shown that for G _p of any large enough cubelike graph G the threshold to have a 1-factor is the same as the threshold to have no isolated vertices. This generalizes a conjecture of K. Weber.     

On random subgraphs of Kneser graphs and their generalizations

A series of results are obtained on the stability of the independence number of random subgraphs of distance graphs, which are natural generalizations of the classical Kneser graphs.     

On the threshold for k-regular subgraphs of random graphs

The k-core of a graph is the largest subgraph of minimum degree at least k. We show that for k sufficiently large, the threshold for the appearance of a k-regular subgraph in the Erdős-Rényi random graph model G(n,p) is at most the threshold for the appearance of a nonempty (k+2)-core. In particular, this pins down the point of appearance of a k-regular subgraph to a window for p of width roughly 2/n for large n and moderately large k. The result is proved by using Tutte’s necessary and sufficient condition for a graph to have a k-factor.     

On distance subgraphs of graphs in spaces of lower dimensions

This paper studies the problem on the frequency of the event where a graph with a fixed number of vertices contains induced subgraphs that are isomorphic to distance graphs in spaces of definite dimensions. In particular, the case of plane and three-dimensional spaces is considered.     

Independence numbers of graphs and generators of ideals

This article investigates the generators of certain homogeneous ideals which are associated with graphs with bounded independence numbers. These ideals first appeared in the theory oft-designs. The main theorem suggests a new approach to the Clique Problem which isNP-complete. This theorem has a more general form in commutative algebra dealing with ideals associated with unions of linear varieties. This general theorem is stated in the article; a corollary to it generalizes Turán’s theorem on the maximum graphs with a prescribed clique number. Research supported in part by NSF Grant MCS77-03533.     

Distribution of the number of spanning regular subgraphs in random graphs

In this paper, we examine the moments of the number of d ‐factors in begin{align*}mathcal{ G}(n,p)end{align*} for all p and d satisfying d³ = o(p²n). We also determine the limiting distribution of the number of d ‐factors inside this range with further restriction that begin{align*}(1-p)sqrt{dn}toinftyend{align*} as n →∞.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Small subgraphs in random graphs and the power of multiple choices

The standard paradigm for online power of two choices problems in random graphs is the Achlioptas process. Here we consider the following natural generalization: Starting with G₀ as the empty graph on n vertices, in every step a set of r edges is drawn uniformly at random from all edges that have not been drawn in previous steps. From these, one edge has to be selected, and the remaining r−1 edges are discarded. Thus after N steps, we have seen rN edges, and selected exactly N out of these to create a graph GN.In a recent paper by Krivelevich, Loh, and Sudakov (2009) [11], the problem of avoiding a copy of some fixed graph F in GN for as long as possible is considered, and a threshold result is derived for some special cases. Moreover, the authors conjecture a general threshold formula for arbitrary graphs F. In this work we disprove this conjecture and give the complete solution of the problem by deriving explicit threshold functions N₀(F,r,n) for arbitrary graphs F and any fixed integer r. That is, we propose an edge selection strategy that a.a.s. (asymptotically almost surely, i.e. with probability 1−o(1) as n→∞) avoids creating a copy of F for as long as N=o(N₀), and prove that any online strategy will a.a.s. create such a copy once N=ω(N₀).     

On chromatic numbers of nearly Kneser distance graphs

A family of distance graphs whose structure is close to that of Kneser graphs is studied. New lower and upper bounds for the chromatic numbers of such graphs are obtained, and relations between these numbers are considered. The structure of certain important independent sets of the family of graphs under consideration is described, and the cardinalities of these sets are explicitly calculated.