The n‐ordered graphs: A new graph class

For a positive integer n, we introduce the new graph class of n‐ordered graphs, which generalize partial n‐trees. Several characterizations are given for the finite n‐ordered graphs, including one via a combinatorial game. We introduce new countably infinite graphs R⁽ⁿ⁾, which we name the infinite random n‐ordered graphs. The graphs R⁽ⁿ⁾ play a crucial role in the theory of n‐ordered graphs, and are inspired by recent research on the web graph and the infinite random graph. We characterize R⁽ⁿ⁾ as a limit of a random process, and via an adjacency property and a certain folding operation. We prove that the induced subgraphs of R⁽ⁿ⁾ are exactly the countable n‐ordered graphs. We show that all countable groups embed in the automorphism group of R⁽ⁿ⁾. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 204–218, 2009     

New Infinite Families of 2‐Edge‐Balanced Graphs

A graph G of order n is called t‐edge‐balanced if G satisfies the property that there exists a positive λ for which every graph of order n and size t is contained in exactly λ distinct subgraphs of isomorphic to G. We call λ the index of G. In this article, we obtain new infinite families of 2‐edge‐balanced graphs.     

Small maximal matchings of random cubic graphs

We consider the expected size of a smallest maximal matching of cubic graphs. Firstly, we present a randomized greedy algorithm for finding a small maximal matching of cubic graphs. We analyze the average‐case performance of this heuristic on random n‐vertex cubic graphs using differential equations. In this way, we prove that the expected size of the maximal matching returned by the algorithm is asymptotically almost surely (a.a.s.) less than 0.34623n. We also give an existence proof which shows that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. less than 0.3214n. It is known that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. larger than 0.3158n. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 293–323, 2009     

New ‐Edge‐Balanced Graphs from Bipartite Graphs

Let G be a graph of order n satisfying that there exists for which every graph of order n and size t is contained in exactly λ distinct subgraphs of the complete graph isomorphic to G. Then G is called t‐edge‐balanced and λ the index of G. In this article, new examples of 2‐edge‐balanced graphs are constructed from bipartite graphs and some further methods are introduced to obtain more from old.     

Sparse graphs: Metrics and random models

Recently, Bollobás, Janson and Riordan introduced a family of random graph models producing inhomogeneous graphs with n vertices and Θ(n) edges whose distribution is characterized by a kernel, i.e., a symmetric measurable function κ: [0, 1]² → [0, ∞). To understand these models, we should like to know when different kernels κ give rise to “similar” graphs, and, given a real‐world network, how “similar” is it to a typical graph G(n, κ) derived from a given kernel κ. The analogous questions for dense graphs, with Θ(n²) edges, are answered by recent results of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi, who showed that several natural metrics on graphs are equivalent, and moreover that any sequence of graphs converges in each metric to a graphon, i.e., a kernel taking values in [0, 1]. Possible generalizations of these results to graphs with o(n²) but ω(n) edges are discussed in a companion article [Bollobás and Riordan, London Math Soc Lecture Note Series 365 (2009), 211–287]; here we focus only on graphs with Θ(n) edges, which turn out to be much harder to handle. Many new phenomena occur, and there are a host of plausible metrics to consider; many of these metrics suggest new random graph models and vice versa. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 1‐38, 2011     

Asymptotic enumeration of sparse 2‐connected graphs

We determine an asymptotic formula for the number of labelled 2‐connected (simple) graphs on n vertices and m edges, provided that m ‐ n →∞ and m = O(nlog n) as n →∞. This is the entire range of m not covered by previous results. The proof involves determining properties of the core and kernel of random graphs with minimum degree at least 2. The case of 2‐edge‐connectedness is treated similarly. We also obtain formulae for the number of 2‐connected graphs with given degree sequence for most (“typical”) sequences. Our main result solves a problem of Wright from 1983. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Critical graphs for subpancyclicity of 3‐connected claw‐free graphs

Let be the family of graphs G such that all sufficiently large k ‐connected claw‐free graphs which contain no induced copies of G are subpancyclic. We show that for every k≥3 the family is infinite and make the first step toward the complete characterization of the family . © 2009 Wiley Periodicals, Inc. J Graph Theory 62, 263–278, 2009     

Planar graphs: Random walks and bipartiteness testing

We initiates the study of property testing in arbitrary planar graphs. We prove that bipartiteness can be tested in constant time, improving on the previous bound of for graphs on n vertices. The constant‐time testability was only known for planar graphs with bounded degree. Our algorithm is based on random walks. Since planar graphs have good separators, that is, bad expansion, our analysis diverges from standard techniques that involve the fast convergence of random walks on expanders. We reduce the problem to the task of detecting an odd‐parity cycle in a multigraph induced by constant‐length cycles. We iteratively reduce the length of cycles while preserving the detection probability, until the multigraph collapses to a collection of easily discoverable self‐loops. Our approach extends to arbitrary minor‐free graphs. We also believe that our techniques will find applications to testing other properties in arbitrary minor‐free graphs.     

Quasi‐randomness of graph balanced cut properties

Quasi‐random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi‐randomness of graphs. Let k ≥ 2 be a fixed integer, α₁,…,α_k be positive reals satisfying \begin{align*}\sum_{i} \alpha_i = 1\end{align*} and (α₁,…,α_k)≠(1/k,…,1/k), and G be a graph on n vertices. If for every partition of the vertices of G into sets V ₁,…,V _k of size α₁n,…,α_kn, the number of complete graphs on k vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi‐random. However, the method of quasi‐random hypergraphs they used did not provide enough information to resolve the case (1/k,…,1/k) for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi‐random. Janson also posed the same question in his study of quasi‐randomness under the framework of graph limits. In this paper, we positively answer their question. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Partial characterizations of circular‐arc graphs

A circular‐arc graph is the intersection graph of a family of arcs on a circle. A characterization by forbidden induced subgraphs for this class of graphs is not known, and in this work we present a partial result in this direction. We characterize circular‐arc graphs by a list of minimal forbidden induced subgraphs when the graph belongs to any of the following classes: P₄ ‐free graphs, paw‐free graphs, claw‐free chordal graphs and diamond‐free graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 289–306, 2009     

The phase transition in inhomogeneous random graphs

The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power‐law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real‐world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007     

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     

Existential closure of block intersection graphs of infinite designs having finite block size and index

In this article we study the n‐existential closure property of the block intersection graphs of infinite t‐(v, k, λ) designs for which the block size k and the index λ are both finite. We show that such block intersection graphs are 2‐e.c. when 2?t?k ? 1. When λ = 1 and 2?t?k, then a necessary and sufficient condition on n for the block intersection graph to be n‐e.c. is that n?min{t, ?(k ? 1)/(t ? 1)? + 1}. If λ?2 then we show that the block intersection graph is not n‐e.c. for any n?min{t + 1, ?k/t? + 1}, and that for 3?n?min{t, ?k/t?} the block intersection graph is potentially but not necessarily n‐e.c. The cases t = 1 and t = k are also discussed. © 2011 Wiley Periodicals, Inc. J Combin Designs 19: 85–94, 2011     

Sparse pseudo‐random graphs are Hamiltonian

In this article we study Hamilton cycles in sparse pseudo‐random graphs. We prove that if the second largest absolute value λ of an eigenvalue of a d‐regular graph G on n vertices satisfies and n is large enough, then G is Hamiltonian. We also show how our main result can be used to prove that for every c >0 and large enough n a Cayley graph X (G,S), formed by choosing a set S of c log⁵ n random generators in a group G of order n, is almost surely Hamiltonian. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 17–33, 2003     

s‐Regular cubic graphs as coverings of the complete bipartite graph K3,3

A graph is s‐regular if its automorphism group acts freely and transitively on the set of s‐arcs. An infinite family of cubic 1‐regular graphs was constructed in [10], as cyclic coverings of the three‐dimensional Hypercube. In this paper, we classify the s‐regular cyclic coverings of the complete bipartite graph K_3,3 for each ≥ 1 whose fibre‐preserving automorphism subgroups act arc‐transitively. As a result, a new infinite family of cubic 1‐regular graphs is constructed. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 101–112, 2004     

Thickness‐two graphs part one: New nine‐critical graphs,permuted layer graphs,and Catlin's graphs

The purpose of this article is to offer new insight and tools toward the pursuit of the largest chromatic number in the class of thicknesstwo graphs. At present, the highest chromatic number known for a thickness‐two graph is 9, and there is only one known color‐critical such graph. We introduce 40 small 9‐critical thickness‐two graphs, and then use a newconstruction, the permuted layer graphs, together with a construction of Hajós to create an infinite family of 9‐critical thickness‐two graphs. Finally, a non‐trivial infinite subfamily of Catlin's graphs, with directly computable chromatic numbers, is shown to have thickness two. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 198–214, 2008     

Merging percolation on Zd and classical random graphs: Phase transition

We study a random graph model which is a superposition of bond percolation on Z^d with parameter p, and a classical random graph G(n,c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so‐called “rank 1 case” of inhomogeneous random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters c ≥ 0 and 0 ≤ p < p_c, where p_c = p_c(d) is the critical probability for the bond percolation on Z^d. The phase transition is of second order as in the classical random graph. We find the scaled size of the largest connected component in the supercritical regime. We also provide a sharp upper bound for the largest connected component in the subcritical regime. The latter is a new result for inhomogeneous random graphs with unbounded kernels. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Restricted b‐factors in bipartite graphs and t‐designs

We present a new equivalence result between restricted b‐factors in bipartite graphs and combinatorial t‐designs. This result is useful in the construction of t‐designs by polyhedral methods. We propose a novel linear integer programming formulation, which we call GDP, for the problem of finding t‐designs that has a noteworthy advantage compared to the traditional set‐covering formulation. We analyze some polyhedral properties of GPD, implement a branch‐and‐cut algorithm using it and solve several instances of small designs to compare with another point‐block formulation found in the literature. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 169–182, 2006     

Planar embeddings with infinite faces

We investigate vertex‐transitive graphs that admit planar embeddings having infinite faces, i.e., faces whose boundary is a double ray. In the case of graphs with connectivity exactly 2, we present examples wherein no face is finite. In particular, the planar embeddings of the Cartesian product of the r‐valent tree with K₂ are comprehensively studied and enumerated, as are the automorphisms of the resulting maps, and it is shown for r = 3 that no vertex‐transitive group of graph automorphisms is extendable to a group of homeomorphisms of the plane. We present all known families of infinite, locally finite, vertex‐transitive graphs of connectivity 3 and an infinite family of 4‐connected graphs that admit planar embeddings wherein each vertex is incident with an infinite face. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 257–275, 2003