We show that if G is a simple outerplanar graph and H is a graph with the same Tutte polynomial as G, then H is also outerplanar. Examples show that the condition of G being simple cannot be omitted.

Two players share a connected graph with non-negative weights on the vertices. They alternately take the vertices (one in each turn) and collect their weights. The rule they have to obey is that the remaining part of the graph must be connected after each move. We conjecture that the first player can get at least half of the weight of any tree with an even number of vertices. We provide a strategy for the first player to get at least 1/4 of an even tree. Moreover, we confirm the conjecture for subdivided stars. The parity condition is necessary: Alice gets nothing on a three-vertex path with all the weight at the middle. We suspect a kind of general parity phenomenon, namely, that the first player can gather a substantial portion of the weight of any ‘simple enough’ graph with an even number of vertices.

A vertex i of a graph G = (V,E) is said to be controlled by $M \subseteq V$ if the majority of the elements ofthe neighborhood of i (including itself) belong to M. The setM is a monopoly in G if every vertex $i\in V$ iscontrolled by M. Given a set $M \subseteq V$ and two graphsG1 = ($V,E_1$) and G2 = ($V,E_2$) where $E_1\subseteq E_2$, themonopoly verification problem (mvp) consists of decidingwhether there exists a sandwich graph G = (V,E) (i.e., a graphwhere $E_1\subseteq E\subseteq E_2$) such that M is a monopolyin G = (V,E). If the answer to the mvp is No, we thenconsider the max-controlled set problem (mcsp), whoseobjective is to find a sandwich graph G = (V,E) such that thenumber of vertices of G controlled by M is maximized. The mvp can be solved in polynomial time; the mcsp, however, isNP-hard. In this work, we present a deterministic polynomial timeapproximation algorithm for the mcsp with ratio$\frac{1}{2}$ + $\frac{1+\sqrt{n}}{2n-2}$, where n=|V|>4. (Thecase $n\leq4$ is solved exactly by considering the parameterizedversion of the mcsp.) The algorithm is obtained through theuse of randomized rounding and derandomization techniques based onthe method of conditional expectations. Additionally, we show howto improve this ratio if good estimates of expectation are obtained in advance.

Dubhashi, Jonasson and Ranjan Dubhashi, Jonasson and Ranjan (2007) study the negative dependence properties of Srinivasan's sampling processes (SSPs), random processes which sample sets of a fixed size with prescribed marginals. In particular they prove that linear SSPs have conditional negative association, by using the Feder–Mihail theorem and a coupling argument. We consider a broader class of SSPs that we call tournament SSPs (TSSPs). These have a tree-like structure and we prove that they have conditional negative association. Our approach is completely different from that of Dubhashi, Jonasson and Ranjan. We give an abstract characterization of TSSPs, and use this to deduce that certain conditioned TSSPs are themselves TSSPs. We show that TSSPs have negative association, and hence conditional negative association. We also give an example of an SSP that does not have negative association.

We show that a set A ⊂ {0, 1}n with edge-boundary of size at mostcan be made into a subcube by at most (2ε/log2(1/ε))|A| additions and deletions, provided ε is less than an absolute constant.

We deduce that if A ⊂ {0, 1}n has size 2t for some t ∈ ℕ, and cannot be made into a subcube by fewer than δ|A| additions and deletions, then its edge-boundary has size at leastprovided δ is less than an absolute constant. This is sharp whenever δ = 1/2j for some j ∈ {1, 2, . . ., t}.

The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn, Kim, Lovász and Vu [25] yielded a (log logn)2 polynomial time approximation. We refine the upper bound of [25], and show that the resulting bound is sharp and explicitly computable in random graphs. Cooper and Frieze showed that the cover time of the largest component of the Erdős–Rényi random graph G(n, c/n) in the supercritical regime with c > 1 fixed, is asymptotic to ϕ(c)nlog2n, where ϕ(c) → 1 as c ↓ 1. However, our new bound implies that the cover time for the critical Erdős–Rényi random graph G(n, 1/n) has order n, and shows how the cover time evolves from the critical window to the supercritical phase. Our general estimate also yields the order of the cover time for a variety of other concrete graphs, including critical percolation clusters on the Hamming hypercube {0, 1}n, on high-girth expanders, and on tori ℤdn for fixed large d. This approach also gives a simpler proof of a result of Aldous [2] that the cover time of a uniform labelled tree on k vertices is of order k3/2. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman [45], is within a constant factor of the cover time. Finally, we prove that for any connected graph, adding an edge can increase the cover time by at most a factor of 4.

We introduce a discrete random process which we call the passenger model, and show that it is connected to a certain random model of the assignment problem and in particular to the so-called Buck–Chan–Robbins urn process. We propose a conjecture on the distribution of the location of the minimum cost assignment in a cost matrix with zeros at specified positions and remaining entries of exponential distribution. The conjecture is consistent with earlier results on the participation probability of an individual matrix entry. We also use the passenger model to verify a conjecture by V. Dotsenko on the assignment problem.

An r-cut of the complete r-uniform hypergraph Krn is obtained by partitioning its vertex set into r parts and taking all edges that meet every part in exactly one vertex. In other words it is the edge set of a spanning complete r-partite subhypergraph of Krn. An r-cut cover is a collection of r-cuts such that each edge of Krn is in at least one of the cuts. While in the graph case r = 2 any 2-cut cover on average covers each edge at least 2-o(1) times, when r is odd we exhibit an r-cut cover in which each edge is covered exactly once. When r is even no such decomposition can exist, but we can bound the average number of times an edge is cut in an r-cut cover between and . The upper bound construction can be reformulated in terms of a natural polyhedral problem or as a probability problem, and we solve the latter asymptotically.

Szemerédi's Regularity Lemma is an important tool for analysing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we prove a sparse Regularity Lemma that holds for all graphs. More generally, we give a Regularity Lemma that holds for arbitrary real matrices.

In this paper we study planar first-passage percolation (FPP) models on random Delaunay triangulations. In [14], Vahidi-Asl and Wierman showed, using sub-additivity theory, that the rescaled first-passage time converges to a finite and non-negative constant μ. We show a sufficient condition to ensure that μ>0 and derive some upper bounds for fluctuations. Our proofs are based on percolation ideas and on the method of martingales with bounded increments.

Let P be a set of n points in ℝ3, and let k ≤ n be an integer. A sphere σ is k-rich with respect to P if |σ ∩ P| ≥ k, and is η-non-degenerate, for a fixed fraction 0 < η < 1, if no circle γ ⊂ σ contains more than η|σ ∩ P| points of P.

We improve the previous bound given in [1] on the number of k-rich η-non-degenerate spheres in 3-space with respect to any set of n points in ℝ3, from O(n4/k5 + n3/k3), which holds for all 0 < η < 1/2, to O*(n4/k11/2 + n2/k2), which holds for all 0 < η < 1 (in both bounds, the constants of proportionality depend on η). The new bound implies the improved upper bound O*(n58/27) ≈ O(n2.1482) on the number of mutually similar triangles spanned by n points in ℝ3; the previous bound was O(n13/6) ≈ O(n2.1667) [1].

Extending an old conjecture of Tutte, Jaeger conjectured in 1988 that for any fixed integer p ≥ 1, the edges of any 4p-edge connected graph can be oriented so that the difference between the outdegree and the indegree of each vertex is divisible by 2p+1. It is known that it suffices to prove this conjecture for (4p+1)-regular, 4p-edge connected graphs. Here we show that there exists a finite p0 such that for every p > p0 the assertion of the conjecture holds for all (4p+1)-regular graphs that satisfy some mild quasi-random properties, namely, the absolute value of each of their non-trivial eigenvalues is at most c1p2/3 and the neighbourhood of each vertex contains at most c2p3/2 edges, where c1, c2 > 0 are two absolute constants. In particular, this implies that for p > p0 the assertion of the conjecture holds asymptotically almost surely for random (4p+1)-regular graphs.

Let d = (d1, d2, . . ., dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph.

Although there are many results of this kind, they are restricted to the sparse case with only a few exceptions. Our focus is instead on the case where the average degree is approximately a constant fraction of n.

Our approach is the multidimensional saddle-point method. This extends the enumerative work of McKay and Wormald (1990) and is analogous to the theory developed for bipartite graphs by Greenhill and McKay (2009).

The study of the structural properties of large random planar graphs has become in recent years a field of intense research in computer science and discrete mathematics. Nowadays, a random planar graph is an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints.

In this paper we focus on the structure of random 2-connected planar graphs regarding the sizes of their 3-connected building blocks, which we call cores. In fact, we prove a general theorem regarding random biconnected graphs from various classes. If Bn is a graph drawn uniformly at random from a suitable class of labelled biconnected graphs, then we show that with probability 1 − o(1) as n → ∞, Bn belongs to exactly one of the following categories:

1. (i) either there is a unique giant core in Bn, that is, there is a 0 < c = c() < 1 such that the largest core contains ~ cn vertices, and every other core contains at most nα vertices, where 0 < α = α() < 1;

2. (ii) or all cores of Bn contain O(logn) vertices.