A Harris‐Kesten theorem for confetti percolation

Percolation properties of the dead leaves model, also known as confetti percolation, are considered. More precisely, we prove that the critical probability for confetti percolation with square‐shaped leaves is 1/2. This result is related to a question of Benjamini and Schramm concerning disk‐shaped leaves and can be seen as a variant of the Harris‐Kesten theorem for bond percolation. The proof is based on techniques developed by Bollobás and Riordan to determine the critical probability for Voronoi and Johnson‐Mehl percolation. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 361–385, 2015     

Small subgraphs in preferential attachment networks

In this paper, we give an overview of old and new results on the global and average local clustering coefficients as well as on the distribution of small subgraphs in the Bollobás–Riordan and Buckley–Osthus models of web graphs.     

Interlacement and activities in delta-matroids

We generalize theories of graph, matroid, and ribbon-graph activities to delta-matroids. As a result, we obtain an activity based feasible-set expansion for a transition polynomial of delta-matroids defined by Brijder and Hoogeboom. This result yields feasible-set expansions for the two-variable Bollobás–Riordan and interlace polynomials of a delta-matroid. In the former case, the expansion obtained directly generalizes the activities expansions of the Tutte polynomial of graphs and matroids.     

A Tutte polynomial for maps II: The non-orientable case

We construct a new polynomial invariant of maps (graphs embedded in a closed surface, orientable or non-orientable), which contains as specializations the Krushkal polynomial, the Bollobás—Riordan polynomial, the Las Vergnas polynomial, and their extensions to non-orientable surfaces, and hence in particular the Tutte polynomial. Other evaluations include the number of local flows and local tensions taking non-identity values in a given finite group.     

Limits of locally–globally convergent graph sequences

The colored neighborhood metric for sparse graphs was introduced by Bollobás and Riordan [BR11]. The corresponding convergence notion refines a convergence notion introduced by Benjamini and Schramm [BS01]. We prove that even in this refined sense, the limit of a convergent graph sequence (with uniformly bounded degree) can be represented by a graphing. We study various topics related to this convergence notion such as: Bernoulli graphings, factor of i.i.d. processes and hyperfiniteness.     

Topological Graph Polynomial and Quantum Field Theory Part II: Mehler Kernel Theories

We define a new topological polynomial extending the Bollobás–Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behaviour under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse–Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided.     

Percolation on random Johnson–Mehl tessellations and related models

We make use of the recent proof that the critical probability for percolation on random Voronoi tessellations is 1/2 to prove the corresponding result for random Johnson–Mehl tessellations, as well as for two-dimensional slices of higher-dimensional Voronoi tessellations. Surprisingly, the proof is a little simpler for these more complicated models. B. Bollobás’s research was supported in part by NSF grants CCR-0225610 and DMS-0505550 and ARO grant W911NF-06-1-0076. O. Riordan’s research was supported by a Royal Society Research Fellowship.     

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     

The Las Vergnas polynomial for embedded graphs

The Las Vergnas polynomial is an extension of the Tutte polynomial to cellularly embedded graphs. It was introduced by Michel Las Vergnas in 1978 as special case of his Tutte polynomial of a morphism of matroids. While the general Tutte polynomial of a morphism of matroids has a complete set of deletion–contraction relations, its specialisation to cellularly embedded graphs does not. Here we extend the Las Vergnas polynomial to graphs in pseudo-surfaces. We show that in this setting we can define deletion and contraction for embedded graphs consistently with the deletion and contraction of the underlying matroid perspective, thus yielding a version of the Las Vergnas polynomial with complete recursive definition. This also enables us to obtain a deeper understanding of the relationships among the Las Vergnas polynomial, the Bollobás–Riordan polynomial, and the Krushkal polynomial. We also take this opportunity to extend some of Las Vergnas’ results on Eulerian circuits from graphs in surfaces of low genus to graphs in surfaces of arbitrary genus.     

Invariants of Composite Networks Arising as a Tensor Product

We generalize Brylawski’s formula of the Tutte polynomial of a tensor product of matroids to colored connected graphs, matroids, and disconnected graphs. Unlike the non-colored tensor product where all edges have to be replaced by the same graph, our colored generalization of the tensor product operation allows individual edge replacement. The colored Tutte polynomials we compute exists by the results of Bollobás and Riordan. The proof depends on finding the correct generalization of the two components of the pointed Tutte polynomial, first studied by Brylawski and Oxley, and on careful enumeration of the connected components in a tensor product. Our results make the calculation of certain invariants of many composite networks easier, provided that the invariants are obtained from the colored Tutte polynomials via substitution and the composite networks are represented as tensor products of colored graphs. In particular, our method can be used to calculate (with relative ease) the expected number of connected components after an accident hits a composite network in which some major links are identical subnetworks in themselves.      

List multicolorings of graphs with measurable sets

In an ordinary list multicoloring of a graph, the vertices are “colored” with subsets of pre‐assigned finite sets (called “lists”) in such a way that adjacent vertices are colored with disjoint sets. Here we consider the analog of such colorings in which the lists are measurable sets from an arbitrary atomless, semifinite measure space, and the color sets are to have prescribed measures rather than prescribed cardinalities. We adapt a proof technique of Bollobás and Varopoulos to prove an analog of one of the major theorems about ordinary list multicolorings, a generalization and extension of Hall's marriage theorem, due to Cropper, Gyárfás, and Lehel. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 179–193, 2007     

On the limit of large girth graph sequences

Let d≥2 be given and let μ be an involution-invariant probability measure on the space of trees T ∈ T _d with maximum degrees at most d. Then μ arises as the local limit of some sequence {G _n } _n=1^∞ of graphs with all degrees at most d. This answers Question 3.3 of Bollobás and Riordan [4].     

Some graphs with chromatic number three

A question of P. Erdös is solved by showing that certain graphs have chromatic number at most three. The proof proceeds by showing a conjecture of Erdös and Bollobás holds, namely, that under certain circumstances, a graph which contains an odd circuit must contain an odd circuit with diagonal.     

The Bishop–Phelps–Bollobás and approximate hyperplane series properties

We study the Bishop–Phelps–Bollobás property for operators between Banach spaces. Sufficient conditions are given for generalized direct sums of Banach spaces with respect to a uniformly monotone Banach sequence lattice to have the approximate hyperplane series property. This result implies that Bishop–Phelps–Bollobás theorem holds for operators from ${?}_1$ into such direct sums of Banach spaces. We also show that the direct sum of two spaces with the approximate hyperplane series property has such property whenever the norm of the direct sum is absolute.     

Properties of intersecting families of ordered sets

The first part of this paper deals with families of ordered k-tuples having a common element in different position. A quite general theorem is given and special cases are considered. The second part deals with t families of sets with some intersection properties, and generalizes results by Bollobás, Frankl, Lovász and Füredi to this case.     

Elementary proof of a counting formula for acyclic bipartite tournaments

This note gives a simple proof of a formula due to Bollobás, Frank and Karoński for counting acyclic bipartit tournaments. © 1995 John Wiley & Sons, Inc.     

Large deviation analysis for layered percolation problems on the complete graph

We analyze the large deviation properties for the (multitype) version of percolation on the complete graph – the simplest substitutive generalization of the Erd&0151;s‐Rènyi random graph that was treated in article by Bollobás et al. (Random Structures Algorithms 31 (2007), 3–122). Here the vertices of the graph are divided into a fixed finite number of sets (called layers) the probability of {u,v} being in our edge set depends on the respective layers of u and v. We determine the exponential rate function for the probability that a giant component occupies a fixed fraction of the graph, while all other components are small. We also determine the exponential rate function for the probability that a particular exploration process on the random graph will discover a certain fraction of vertices in each layer, without encountering a giant component.© 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 460–492, 2012     

Topological Graph Polynomials in Colored Group Field Theory

In this paper, we analyze the open Feynman graphs of the Colored Group Field Theory introduced in Gurau (Colored group field theory, arXiv:0907.2582 [hep-th]). We define the boundary graph ${\mathcal{G}_{\partial}}In this paper, we analyze the open Feynman graphs of the Colored Group Field Theory introduced in Gurau (Colored group field theory, arXiv:0907.2582 [hep-th]). We define the boundary graph G_?{\mathcal{G}_{\partial}} of an open graph G{\mathcal{G}} and prove it is a cellular complex. Using this structure we generalize the topological (Bollobás–Riordan) Tutte polynomials associated to (ribbon) graphs to topological polynomials adapted to Colored Group Field Theory graphs in arbitrary dimension.     

Discrepancy of random graphs and hypergraphs

Answering in a strong form a question posed by Bollobás and Scott, in this paper we determine the discrepancy between two random k‐uniform hypergraphs, up to a constant factor depending solely on k. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 47, 147–162, 2015     

On the minimum number of edges of two-connected graphs with given diameter

We answer the following question: what is the minimum number of edges of a 2-connected graph with a given diameter? This problem stems from survivable telecommunication network design with grade-of-service constraints. In this paper, we prove tight bounds for 2-connected graphs and for 2-edge-connected graphs. The bound for 2-connected graphs was a conjecture of B. Bollobás (AMH 75–80) [3].