Given a graph (Formula presented.), its auxiliary square-graph (Formula presented.) is the graph whose vertices are the non-edges of (Formula presented.) and whose edges are the pairs of non-edges which induce a square (i.e., a 4-cycle) in (Formula presented.). We determine the threshold edge-probability (Formula presented.) at which the Erdős–Rényi random graph (Formula presented.) begins to asymptotically almost surely (a.a.s.) have a square-graph with a connected component whose squares together cover all the vertices of (Formula presented.). We show (Formula presented.), a polylogarithmic improvement on earlier bounds on (Formula presented.) due to Hagen and the authors. As a corollary, we determine the threshold (Formula presented.) at which the random right-angled Coxeter group (Formula presented.) a.a.s. becomes strongly algebraically thick of order 1 and has quadratic divergence.
In this paper we introduce new models of random graphs, arising from Latin squares which include random Cayley graphs as a special case. We investigate some properties of these graphs including their clique, independence and chromatic numbers, their expansion properties as well as their connectivity and Hamiltonicity. The results obtained are compared with other models of random graphs and several similarities and differences are pointed out. For many properties our results for the general case are as strong as the known results for random Cayley graphs and sometimes improve the previously best results for the Cayley case. (C) 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011
Given a group G, the model G(G,p) denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. In this article we show that for any epsilon>0 and any family of groups G(k) of order n(k) for which nk, a graph kG(Gk,p) with high probability has diameter at most 2 if p(2+epsilon)lognknk and with high probability has diameter greater than 2 if p(14+epsilon)lognknk. We also provide examples of families of graphs which show that both of these results are best possible. Of particular interest is that for some families of groups, the corresponding random Cayley graphs achieve Diameter 2 significantly faster than the Erds-Renyi random graphs.
A probability measure on the subsets of the edge set of a graph G is a 1‐independent probability measure (1‐ipm) on G if events determined by edge sets that are at graph distance at least 1 apart in G are independent. Given a 1‐ipm , denote by the associated random graph model. Let denote the collection of 1‐ipms on G for which each edge is included in with probability at least p. For , Balister and Bollobás asked for the value of the least p⋆ such that for all p > p⋆ and all , almost surely contains an infinite component. In this paper, we significantly improve previous lower bounds on p⋆. We also determine the 1‐independent critical probability for the emergence of long paths on the line and ladder lattices. Finally, for finite graphs G we study f1, G(p), the infimum over all of the probability that is connected. We determine f1, G(p) exactly when G is a path, a complete graph and a cycle of length at most 5.
The Alon-Roichman theorem states that for every $\ge > 0$ there is a constant $c(\ge)$, such that the Cayley graph of a finite group $G$ with respect to $c(\ge)\log{\abs{G}}$ elements of $G$, chosen independently and uniformly at random, has expected second largest eigenvalue less than $\ge$. In particular, such a graph is an expander with high probability.
Landau and Russell, and independently Loh and Schulman, improved the bounds of the theorem. Following Landau and Russell we give a simpler proof of the result, improving the bounds even further. When considered for a general group $G$, our bounds are in a sense best possible.
We also give a generalisation of the Alon-Roichman theorem to random coset graphs.
Let denote the power set of [n], ordered by inclusion, and let denote the random poset obtained from by retaining each element from independently at random with probability p and discarding it otherwise. Given any fixed poset F we determine the threshold for the property that contains F as an induced subposet. We also asymptotically determine the number of copies of a fixed poset F in . Finally, we obtain a number of results on the Ramsey properties of the random poset .
In breakthrough results, Saxton-Thomason and Balogh-Morris-Samotij developed powerful theories of hypergraph containers. In this paper, we explore some consequences of these theories. We use a simple container theorem of Saxton-Thomason and an entropy-based framework to deduce container and counting theorems for hereditary properties of k-colorings of very general objects, which include both vertex- and edge-colorings of general hypergraph sequences as special cases. In the case of sequences of complete graphs, we further derive characterization and transference results for hereditary properties in terms of their stability families and extremal entropy. This covers within a unified framework a great variety of combinatorial structures, some of which had not previously been studied via containers: directed graphs, oriented graphs, tournaments, multigraphs with bounded multiplicity, and multicolored graphs among others. Similar results were recently and independently obtained by Terry.
A random graph model on a host graph (Formula presented.) is said to be 1-independent if for every pair of vertex-disjoint subsets (Formula presented.) of (Formula presented.), the state of edges (absent or present) in (Formula presented.) is independent of the state of edges in (Formula presented.). For an infinite connected graph (Formula presented.), the 1-independent critical percolation probability (Formula presented.) is the infimum of the (Formula presented.) such that every 1-independent random graph model on (Formula presented.) in which each edge is present with probability at least (Formula presented.) almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that (Formula presented.) tends to a limit in (Formula presented.) as (Formula presented.), and they asked for the value of this limit. We make progress on a related problem by showing that (Formula presented.) In fact, we show that the equality above remains true if the sequence of complete graphs (Formula presented.) is replaced by a sequence of weakly pseudorandom graphs on (Formula presented.) vertices with average degree (Formula presented.). We conjecture the answer to Balister and Bollobás's question is also (Formula presented.).
The basic random k‐SAT problem is: given a set of n Boolean variables, and m clauses of size k picked uniformly at random from the set of all such clauses on our variables, is the conjunction of these clauses satisfiable? Here we consider a variation of this problem where there is a bias towards variables occurring positive—that is, variables occur negated w.p. 0<p<½ and positive otherwise—and study how the satisfiability threshold depends on p. For p<½ this model breaks many of the symmetries of the original random k‐SAT problem, for example, the distribution of satisfying assignments in the Boolean cube is no longer uniform. For any fixed k, we find the asymptotics of the threshold as p approaches 0 or ½ . The former confirms earlier predictions based on numerical studies and heuristic methods from statistical physics.
The goal of this paper is to show the existence (using probabilistic tools) of configurations of lines, boxes, and points with certain interesting combinatorial properties. (i) First, we construct a family of n lines in ℝ3 whose intersection graph is triangle-free of chromatic number Ω (n1∕15). This improves the previously best known bound Ω (log log n) by Norin, and is also the first construction of a triangle-free intersection graph of simple geometric objects with polynomial chromatic number. (ii) Second, we construct a set of n points in ℝd, whose Delaunay graph with respect to axis-parallel boxes has independence number at most n⋅(log n)−(𝑑−1)∕2+o(1). This extends the planar case considered by Chen, Pach, Szegedy, and Tardos.