Let d ≥ d0 be a sufficiently large constant. An graph G is a d-regular graph over n vertices whose second-largest (in absolute value) eigenvalue is at most . For any 0<p<1, Gp is the graph induced by retaining each edge of G with probability p. It is known that for the graph Gp almost surely contains a unique giant component (a connected component with linear number vertices). We show that for the giant component of Gp almost surely has an edge expansion of at least .

We show that the sampling formula induced from a Λ-coalescent process with multiple collisions is regenerative if and only if the measure Λ is either concentrated in 0 (Kingman case) or concentrated in 1 (star-shaped case). The Ewens sampling formula is the only sampling formula in this class which also belongs to Pitman's two-parameter family of sampling distributions.

Consider two graphs G1 and G2 on the same vertex set V and suppose that Gi has mi edges. Then there is a bipartition of V into two classes A and B so that, for both i = 1, 2, we have . This gives an approximate answer to a question of Bollobás and Scott. We also prove results about partitions into more than two vertex classes. Our proofs yield polynomial algorithms.

We introduce the minor-closed, dual-closed class of multi-path matroids. We give a polynomial-time algorithm for computing the Tutte polynomial of a multi-path matroid, we describe their basis activities, and we prove some basic structural properties. Key elements of this work are two complementary perspectives we develop for these matroids: on the one hand, multi-path matroids are transversal matroids that have special types of presentations; on the other hand, the bases of multi-path matroids can be viewed as sets of lattice paths in certain planar diagrams.

The partition functions of the Ising and Potts models in statistical mechanics are well known to be partial evaluations of the Tutte–Whitney polynomial of the appropriate graph. The Ashkin–Teller model generalizes the Ising model and the four-state Potts model, and has been extensively studied since its introduction in 1943. However, its partition function (even in the symmetric case) is not a partial evaluation of the Tutte–Whitney polynomial. In this paper, we show that the symmetric Ashkin–Teller partition function can be obtained from a generalized Tutte–Whitney function which is intermediate in a precise sense between the usual Tutte–Whitney polynomialof the graph and that of its dual.

Consider the following communication problem, which leads to a new notion of edge colouring. The communication network is represented by a bipartite multigraph, where the nodes on one side are the transmitters and the nodes on the other side are the receivers. The edges correspond to messages, and every edge e is associated with an integer c(e), corresponding to the time it takes the message to reach its destination. A proper k-edge-colouring with delays is a function f from the edges to {0, 1, . . ., k − 1}, such that, for every two edges e1 and e2 with the same transmitter, f(e1) ≠ f(e2), and for every two edges e1 and e2 with the same receiver, f(e1) + c(e1) ≢ f(e2) + c(e2) (mod k). Ross, Bambos, Kumaran, Saniee and Widjaja [18] conjectured that there always exists a proper edge colouring with delays using k = Δ + o(Δ) colours, where Δ is the maximum degree of the graph. Haxell, Wilfong and Winkler [11] conjectured that a stronger result holds: k = Δ + 1 colours always suffice. We prove the weaker conjecture for simple bipartite graphs, using a probabilistic approach, and further show that the stronger conjectureholds for some multigraphs, applying algebraic tools.

We show that a random 4-regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. The proof uses an efficient algorithm which a.a.s. 3-colours a random 4-regular graph. The analysis includes use of the differential equation method, and exponential bounds on the tail of random variables associated with branching processes. A substantial part of the analysis applies to random d-regular graphs in general.

Let A be a set of N matrices. Let g(A) ≔ |A + A| + |A · A|, where A + A = {a1 + a2 ∣ ai ∈ A} and A · A = {a1a2 ∣ ai ∈ A} are the sum set and product set. We prove that if the determinant of the difference of any two distinct matrices in A is nonzero, then g(A) cannot be bounded below by cN for any constant c. We also prove that if A is a set of d × d symmetric matrices, then there exists ϵ = ϵ(d)>0 such that g(A)>N1+ϵ. For the first result, we use the bound on the number of factorizations in a generalized progression. For the symmetric case, we use a technical proposition which provides an affine space V containing a large subset E of A, with the property that if an algebraic property holds for a large subset of E, then it holds for V. Then we show that the system a2 : a ∈ V is commutative, allowing us to decompose as eigenspaces simultaneously, so we can finish the proof with induction and a variant of the Erdős–Szemerédi argument.

We describe how non-crossing partitions arise in substitution method calculations. By using efficient algorithms for computing non-crossing partitions we are able to substantially reduce the computational effort, which enables us to compute improved bounds on the percolation thresholds for three percolation models. For the Kagomé bond model we improve bounds from 0.5182 ≤ pc ≤ 0.5335 to 0.522197 ≤ pc ≤ 0.526873, improving the range from 0.0153 to 0.004676. For the (3, 122) bond model we improve bounds from 0.7393 ≤ pc ≤ 0.7418 to 0.739773 ≤ pc ≤ 0.741125, improving the range from 0.0025 to 0.001352. We also improve the upper bound for the hexagonal site model, from 0.794717 to 0.743359.

We show that every regular tournament on n vertices has at least n!/(2 + o(1))n Hamiltonian cycles, thus answering a question of Thomassen [17] and providing a partial answer to a question of Friedgut and Kahn [7]. This compares to an upper bound of about O(n0.25n!/2n) for arbitrary tournaments due to Friedgut and Kahn (somewhat improving Alon's bound of O(n0.5n!/2n)). A key ingredient of the proof is a martingale analysis of self-avoiding walks on a regular tournament.

The strong isometric dimension of a graph G is the least number k such that G isometrically embeds into the strong product of k paths. Using Sperner's theorem, the strong isometric dimension of the Hamming graphs K2 □ Kn is determined.

We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and is ‘attacked by an adversary’. At time t, we add a new vertex xt and m random edges incident with xt, where m is constant. The neighbours of xt are chosen with probability proportional to degree. After adding the edges, the adversary is allowed to delete vertices. The only constraint on the adversarial deletions is that the total number of vertices deleted by time n must be no larger than δn, where δ is a constant. We show that if δ is sufficiently small and m is sufficiently large then with high probability at time n the generated graph has a component of size at least n/30.