The diamond‐free process

Let denote the diamond graph, formed by removing an edge from the complete graph K₄. We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of . We show that, with probability tending to 1 as , the final size of the graph produced is . Our analysis also suggests that the graph produced after i edges are added resembles the uniform random graph, with the additional condition that the edges which do not lie on triangles form a random‐looking subgraph. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 513–551, 2014     

Random walks which prefer unvisited edges: Exploring high girth even degree expanders in linear time

Let be a connected graph with vertices. A simple random walk on the vertex set of G is a process, which at each step moves from its current vertex position to a neighbouring vertex chosen uniformly at random. We consider a modified walk which, whenever possible, chooses an unvisited edge for the next transition; and makes a simple random walk otherwise. We call such a walk an edge‐process (or E‐process). The rule used to choose among unvisited edges at any step has no effect on our analysis. One possible method is to choose an unvisited edge uniformly at random, but we impose no such restriction. For the class of connected even degree graphs of constant maximum degree, we bound the vertex cover time of the E‐process in terms of the edge expansion rate of the graph G, as measured by eigenvalue gap of the transition matrix of a simple random walk on G. A vertex v is ?‐good, if any even degree subgraph containing all edges incident with v contains at least ? vertices. A graph G is ?‐good, if every vertex has the ?‐good property. Let G be an even degree ?‐good expander of bounded maximum degree. Any E‐process on G has vertex cover time This is to be compared with the lower bound on the cover time of any connected graph by a weighted random walk. Our result is independent of the rule used to select the order of the unvisited edges, which could, for example, be chosen on‐line by an adversary. As no walk based process can cover an n vertex graph in less than n – 1 steps, the cover time of the E‐process is of optimal order when . With high probability random r‐regular graphs, even, have . Thus the vertex cover time of the E‐process on such graphs is . © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 36–54, 2015     

Rainbow perfect matchings and Hamilton cycles in the random geometric graph

Given a graph on n vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length n visiting each vertex once and with pairwise different colours on the edges. Similarly (for even n) a rainbow perfect matching is a collection of independent edges with pairwise different colours. In this note we show that if we randomly colour the edges of a random geometric graph with sufficiently many colours, then a.a.s. the graph contains a rainbow perfect matching (rainbow Hamilton cycle) if and only if the minimum degree is at least 1 (respectively, at least 2). More precisely, consider n points (i.e. vertices) chosen independently and uniformly at random from the unit d‐dimensional cube for any fixed . Form a sequence of graphs on these n vertices by adding edges one by one between each possible pair of vertices. Edges are added in increasing order of lengths (measured with respect to the norm, for any fixed ). Each time a new edge is added, it receives a random colour chosen uniformly at random and with repetition from a set of colours, where a sufficiently large fixed constant. Then, a.a.s. the first graph in the sequence with minimum degree at least 1 must contain a rainbow perfect matching (for even n), and the first graph with minimum degree at least 2 must contain a rainbow Hamilton cycle. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 587–606, 2017     

The size of the giant high‐order component in random hypergraphs

The phase transition in the size of the giant component in random graphs is one of the most well‐studied phenomena in random graph theory. For hypergraphs, there are many possible generalizations of the notion of a connected component. We consider the following: two j‐sets (sets of j vertices) are j‐connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. A hypergraph is j‐connected if all j‐sets are pairwise j‐connected. In this paper, we determine the asymptotic size of the unique giant j‐connected component in random k‐uniform hypergraphs for any and .     

Rainbow hamilton cycles in random graphs

One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erd?s‐Rényi random graph G_n,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 G_n,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     

An almost linear time algorithm for finding Hamilton cycles in sparse random graphs with minimum degree at least three

We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph . In this model G is drawn uniformly from graphs with vertex set [n], m edges and minimum degree at least three. We focus on the case where m = cn for constant c. If c is sufficiently large then our algorithm runs in time and succeeds w.h.p. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 73–98, 2015     

On MAXCUT in strictly supercritical random graphs,and coloring of random graphs and random tournaments

We use a theorem by Ding, Lubetzky, and Peres describing the structure of the giant component of random graphs in the strictly supercritical regime, in order to determine the typical size of MAXCUT of in terms of ɛ. We then apply this result to prove the following conjecture by Frieze and Pegden. For every , there exists such that w.h.p. is not homomorphic to the cycle on vertices. We also consider the coloring properties of biased random tournaments. A p‐random tournament on n vertices is obtained from the transitive tournament by reversing each edge independently with probability p. We show that for the chromatic number of a p‐random tournament behaves similarly to that of a random graph with the same edge probability. To treat the case we use the aforementioned result on MAXCUT and show that in fact w.h.p. one needs to reverse edges to make it 2‐colorable.     

On rigidity,orientability, and cores of random graphs with sliders

Suppose that you add rigid bars between points in the plane, and suppose that a constant fraction q of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph the threshold value of c for the appearance of a linear‐sized rigid component as a function of q, generalizing results of Kasiviswanathan, Moore, and Theran. We show that this appearance of a giant component undergoes a continuous transition for and a discontinuous transition for . In our proofs, we introduce a generalized notion of orientability interpolating between 1‐ and 2‐orientability, of cores interpolating between the 2‐core and the 3‐core, and of extended cores interpolating between the 2 + 1‐core and the 3 + 2‐core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of Kasiviswanathan, Moore, and Theran about the size of the 3 + 2‐core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest.     

The random connection model: Connectivity,edge lengths,and degree distributions

Consider the random graph whose vertex set is a Poisson point process of intensity n on . Any two vertices are connected by an edge with probability , independently of all other edges, and independent of the other points of . d is the toroidal metric, r > 0 and is non‐increasing and . Under suitable conditions on g, almost surely, the critical parameter M_n for which does not have any isolated nodes satisfies . Let , and θ be the volume of the unit ball in . Then for all , is connected with probability approaching one as . The bound can be seen to be tight for the usual random geometric graph obtained by setting . We also prove some useful results on the asymptotic behavior of the length of the edges and the degree distribution in the connectivity regime. The results in this paper work for connection functions g that are not necessarily compactly supported but satisfy .     

Counting dense connected hypergraphs via the probabilistic method

In 1990 Bender, Canfield, and McKay gave an asymptotic formula for the number of connected graphs on with m edges, whenever and . We give an asymptotic formula for the number of connected r‐uniform hypergraphs on with m edges, whenever is fixed and with , that is, the average degree tends to infinity. This complements recent results of Behrisch, Coja‐Oghlan, and Kang (the case ) and the present authors (the case , ie, “nullity” or “excess” o(n)). The proof is based on probabilistic methods, and in particular on a bivariate local limit theorem for the number of vertices and edges in the largest component of a certain random hypergraph. The arguments are much simpler than in the sparse case; in particular, we can use “smoothing” techniques to directly prove the local limit theorem, without needing to first prove a central limit theorem.     

On the total variation distance between the binomial random graph and the random intersection graph

When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karoński, Scheinerman, and Singer‐Cohen introduced a random intersection graph by taking randomly assigned sets. The random intersection graph has n vertices and sets assigned to the vertices are chosen to be i.i.d. random subsets of a fixed set M of size m where each element of M belongs to each random subset with probability p, independently of all other elements in M. In 2000, Fill, Scheinerman, and Singer‐Cohen showed that the total variation distance between the random graph and the Erdös‐Rényi graph tends to 0 for any if , where is chosen so that the expected numbers of edges in the two graphs are the same. In this paper, it is proved that the total variation distance still tends to 0 for any whenever .     

Universality of random graphs and rainbow embedding

In this paper we show how to use simple partitioning lemmas in order to embed spanning graphs in a typical member of . Let the maximum density of a graph H be the maximum average degree of all the subgraphs of H. First, we show that for , a graph w.h.p. contains copies of all spanning graphs H with maximum degree at most Δ and maximum density at most d. For , this improves a result of Dellamonica, Kohayakawa, Rödl and Rucińcki. Next, we show that if we additionally restrict the spanning graphs to have girth at least 7 then the random graph contains w.h.p. all such graphs for . In particular, if , the random graph therefore contains w.h.p. every spanning tree with maximum degree bounded by Δ. This improves a result of Johannsen, Krivelevich and Samotij. Finally, in the same spirit, we show that for any spanning graph H with constant maximum degree, and for suitable p, if we randomly color the edges of a graph with colors, then w.h.p. there exists a rainbow copy of H in G (that is, a copy of H with all edges colored with distinct colors). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 546–564, 2016     

Inside the critical window for cohomology of random k‐complexes

We prove sharper versions of theorems of Linial–Meshulam and Meshulam–Wallach which describe the behavior for ‐cohomology of a random k‐dimensional simplicial complex within a narrow transition window. In particular, we show that if Y is a random k‐dimensional simplicial complex with each k‐simplex appearing i.i.d. with probability with and fixed, then the dimension of cohomology is asymptotically Poisson distributed with mean . In the k = 2 case we also prove that in an accompanying growth process, with high probability, vanishes exactly at the moment when the last ‐simplex gets covered by a k‐simplex, a higher‐dimensional analogue of a “stopping time” theorem about connectivity of random graphs due to Bollobás and Thomason. Random Struct. Alg., 2015 © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 102–124, 2016     

Concentration and regularization of random graphs

This paper studies how close random graphs are typically to their expectations. We interpret this question through the concentration of the adjacency and Laplacian matrices in the spectral norm. We study inhomogeneous Erdös‐Rényi random graphs on n vertices, where edges form independently and possibly with different probabilities p_ij. Sparse random graphs whose expected degrees are fail to concentrate; the obstruction is caused by vertices with abnormally high and low degrees. We show that concentration can be restored if we regularize the degrees of such vertices, and one can do this in various ways. As an example, let us reweight or remove enough edges to make all degrees bounded above by O(d) where . Then we show that the resulting adjacency matrix concentrates with the optimal rate: . Similarly, if we make all degrees bounded below by d by adding weight d / n to all edges, then the resulting Laplacian concentrates with the optimal rate: . Our approach is based on Grothendieck‐Pietsch factorization, using which we construct a new decomposition of random graphs. We illustrate the concentration results with an application to the community detection problem in the analysis of networks. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 538–561, 2017     

On the threshold for the Maker‐Breaker H‐game

We study the Maker‐Breaker H‐game played on the edge set of the random graph . In this game two players, Maker and Breaker, alternately claim unclaimed edges of , until all edges are claimed. Maker wins if he claims all edges of a copy of a fixed graph H; Breaker wins otherwise. In this paper we show that, with the exception of trees and triangles, the threshold for an H‐game is given by the threshold of the corresponding Ramsey property of with respect to the graph H. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 558–578, 2016     

A threshold for the Maker‐Breaker clique game

We study the Maker‐Breaker k‐clique game played on the edge set of the random graph G(n, p). In this game, two players, Maker and Breaker, alternately claim unclaimed edges of G(n, p), until all the edges are claimed. Maker wins if he claims all the edges of a k‐clique; Breaker wins otherwise. We determine that the threshold for the graph property that Maker can win this game is at , for all k > 3, thus proving a conjecture from Ref. [Stojakovi? and Szabó, Random Struct Algor 26 (2005), 204–223]. More precisely, we conclude that there exist constants such that when the game is Maker's win a.a.s., and when it is Breaker's win a.a.s. For the triangle game, when k = 3, we give a more precise result, describing the hitting time of Maker's win in the random graph process. We show that, with high probability, Maker can win the triangle game exactly at the time when a copy of K₅ with one edge removed appears in the random graph process. As a consequence, we are able to give an expression for the limiting probability of Maker's win in the triangle game played on the edge set of G(n, p). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 318–341, 2014     

The diameter of inhomogeneous random graphs

In this paper, we study the diameter of inhomogeneous random graphs that are induced by irreducible kernels . The kernels we consider act on separable metric spaces and are almost everywhere continuous. We generalize results known for the Erdős–Rényi model G(n, p) for several ranges of p. We find upper and lower bounds for the diameter of in terms of the expansion factor and two explicit constants that depend on the behavior of the kernel over partitions of the metric space.     

Upper bounds on probability thresholds for asymmetric Ramsey properties

Given two graphs G and H, we investigate for which functions the random graph (the binomial random graph on n vertices with edge probability p) satisfies with probability that every red‐blue‐coloring of its edges contains a red copy of G or a blue copy of H. We prove a general upper bound on the threshold for this property under the assumption that the denser of the two graphs satisfies a certain balancedness condition. Our result partially confirms a conjecture by the first author and Kreuter, and together with earlier lower bound results establishes the exact order of magnitude of the threshold for the case in which G and H are complete graphs of arbitrary size. In our proof we present an alternative to the so‐called deletion method, which was introduced by Rödl and Ruciński in their study of symmetric Ramsey properties of random graphs (i.e. the case G = H), and has been used in many proofs of similar results since then.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 1–28, 2014     

And/or trees: A local limit point of view

We present here a new and universal approach for the study of random and/or trees, unifying in one framework many different models, including some novel ones not yet understood in the literature. An and/or tree is a Boolean expression represented in (one of) its tree shapes. Fix an integer k, take a sequence of random (rooted) trees of increasing size, say , and label each of these random trees uniformly at random in order to get a random Boolean expression on k variables. We prove that, under rather weak local conditions on the sequence of random trees , the distribution induced on Boolean functions by this procedure converges as n tends to infinity. In particular, we characterize two different behaviors of this limit distribution depending on the shape of the local limit of : a degenerate case when the local limit has no leaves; and a non‐degenerate case, which we are able to describe in more details under stronger conditions. In this latter case, we provide a relationship between the probability of a given Boolean function and its complexity. The examples covered by this unified framework include trees that interpolate between models with logarithmic typical distances (such as random binary search trees) and other ones with square root typical distances (such as conditioned Galton–Watson trees).     

Expected time complexity of the auction algorithm and the push relabel algorithm for maximum bipartite matching on random graphs

In this paper we analyze the expected time complexity of the auction algorithm for the matching problem on random bipartite graphs. We first prove that if for every non‐maximum matching on graph G there exist an augmenting path with a length of at most 2l + 1 then the auction algorithm converges after N ? l iterations at most. Then, we prove that the expected time complexity of the auction algorithm for bipartite matching on random graphs with edge probability and c > 1 is w.h.p. This time complexity is equal to other augmenting path algorithms such as the HK algorithm. Furthermore, we show that the algorithm can be implemented on parallel machines with processors and shared memory with an expected time complexity of . © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 384–395, 2016