Abstract: We consider relations between thresholds for monotone set properties and simple lower bounds for such thresholds. A motivating example (Conjecture 2): Given an n-vertex graph H, write pE for the least p such that, for each subgraph H' of H, the expected number of copies of H' in G=G(n, p) is at least 1, and pc for that p for which the probability that G contains (a copy of) H is 1/2. Then (conjecture) pc=O(pElog n). Possible connections with discrete isoperimetry are also discussed.

Abstract: For a fixed graph $H$, we define the rainbow Turan number $\ex^*(n,H)$ to be the maximum number of edges in a graph on $n$ vertices that has a proper edge-colouring with no rainbow $H$. Recall that the (ordinary) Turan number $\ex(n,H)$ is the maximum number of edges in a graph on $n$ vertices that does not contain a copy of $H$. For any non-bipartite $H$ we show that $\ex^*(n,H)=(1+o(1))\ex(n,H)$, and if $H$ is colour-critical we show that $\ex^{*}(n,H)=\ex(n,H)$. When $H$ is the complete bipartite graph $K_{s,t}$ with $s \leq t$ we show $\ex^*(n,K_{s,t}) = O(n^{2-1/s})$, which matches the known bounds for $\ex(n,K_{s,t})$ up to a constant. We also study the rainbow Turan problem for even cycles, and in particular prove the bound $\ex^*(n,C_6) = O(n^{4/3})$, which is of the correct order of magnitude.

Abstract: 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 $x_t$ and $m$ random edges incident with $x_t$, where $m$ is constant. The neighbours of $x_t$ 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 $\delta n$, where $\delta$ is a constant. We show that if $\delta$ 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 $n/30$.

Abstract: Szemeredi's regularity lemma for graphs has proved to be a powerful tool with many subsequent applications. The objective of this paper is to extend the techniques developed by Nagle, Skokan, and t...

Abstract: A simple first moment argument shows that in a randomly chosen $k$-SAT formula with $m$ clauses over $n$ boolean variables, the fraction of satisfiable clauses is $1-2^{-k}+o(1)$ as $m/n\rightarrow\infty$ almost surely. In this paper, we deal with the corresponding algorithmic strong refutation problem: given a random $k$-SAT formula, can we find a certificate that the fraction of satisfiable clauses is $1-2^{-k}+o(1)$ in polynomial time? We present heuristics based on spectral techniques that in the case $k=3$ and $m\geq\ln(n)^6n^{3/2}$, and in the case $k=4$ and $m\geq Cn^2$, find such certificates almost surely. In addition, we present heuristics for bounding the independence number (resp. the chromatic number) of random $k$-uniform hypergraphs from above (resp. from below) for $k=3,4$.

Abstract: The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobas and Sorkin in [3] as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors in [5], evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial. Here, using the medial graph of a planar graph, we relate the one-variable vertex-nullity interlace polynomial to the classical Tutte polynomial when x=y, and conclude that, like the Tutte polynomial, it is in general #P-hard to compute. We also show a relation between the two-variable interlace polynomial and the topological Tutte polynomial of Bollobas and Riordan in [13]. We define the γ invariant as the coefficient of x1 in the vertex-nullity interlace polynomial, analogously to the β invariant, which is the coefficientof x1 in the Tutte polynomial. We then turn to distance hereditary graphs, characterized by Bandelt and Mulder in [9] as being constructed by a sequence ofadding pendant and twin vertices, and show that graphs in this class have γ invariant of 2n+1 when n true twins are added intheir construction. We furthermore show that bipartite distance hereditary graphs are exactly the class of graphs with γ invariant 2, just as the series-parallel graphs are exactly the class of graphs with β invariant 1. In addition, we show that a bipartite distance hereditary graph arises precisely as the circle graph of an Euler circuitin the oriented medial graph of a series-parallel graph. From this we conclude that the vertex-nullity interlace polynomial is polynomial time to compute for bipartite distancehereditary graphs, just as the Tutte polynomial is polynomial time to compute for series-parallel graphs.

Abstract: 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.

Abstract: We study negative dependence properties of a sampling process due to Srinivasan to produce distributions on level sets with given marginals. We give a simple proof that the distribution satisfies negative association. We also show that under a linear match schedule it satisfies the stronger condition of conditional negative association via a non-trivial application of the Feder–Mihail theorem. This method involves the notion of a variable of positive influence. We give some results and related counter-examples which might shed some light on its role in a theory of negative dependence.

Abstract: We study a point process describing the asymptotic behaviour of sizes of the largest components of the random graph G(n, p) in the critical window, that is, for p = n−1 + λn−4/3, where λ is a fixed real number. In particular, we show that this point process has a surprising rigidity. Fluctuations in the large values will be balanced by opposite fluctuations in the small values such that the sum of the values larger than a small (a scaled version of the number of vertices in components of size greater than en2/3) is almost constant.

Abstract: We solve Conway's Angel Problem by showing that the Angel of power 2 has a winning strategy. An old observation of Conway is that we may suppose without loss of generality that the Angel never jumps to a square where he could have already landed at a previous time. We turn this observation around and prove that we may suppose without loss of generality that the Devil never eats a square where the Angel could have already jumped. Then we give a simple winning strategy for the Angel.

Abstract: We show that in the game of angel and devil, played on the planar integer lattice, the angel of power 4 can evade the devil. This answers a question of Berlekamp, Conway and Guy. Independent proofs that work for the angel of power 2 have been given by Kloster and by Mathe.

Abstract: Let G1 and G2 be graphs of order n with maximum degree Δ1 and Δ2, respectively. G1 and G2 are said to pack if there exist injective mappings of the vertex sets into [n], such that the images of the edge sets do not intersect. Sauer and Spencer showed that if Δ1Δ2 < n/2, then G1 and G2 pack. We extend this result by showing that if Δ1Δ2 < n/2, then G1 and G2 do not pack if and only if one of G1 or G2 is a perfect matching and the other either is Kn/2·n/2 with n/2 odd or contains Kn/2 + 1.

Abstract: 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.

Abstract: A simple explicit construction is provided of a partition-valued fragmentation process whose distribution on partitions of [n] = 1,. . .,n at time θ ≥ 0 is governed by the Ewens sampling formula with parameter θ. These partition-valued processes are exchangeable and consistent, as n varies. They can be derived by uniform sampling from a corresponding mass fragmentation process defined by cutting a unit interval at the points of a Poisson process with intensity θx−1dx on/mathbbR+, arranged to beintensifying as θ increases.

Abstract: Consider two graphs $G_1$ and $G_2$ on the same vertex set $V$ and suppose that $G_i$ has $m_i$ edges. Then there is a bipartition of $V$ into two classes $A$ and $B$ so that, for both $i=1,2$, we have $e_{G_i}(A,B) \geq m_i/2-\sqrt{m_i}$. This gives an approximate answer to a question of Bollobas and Scott. We also prove results about partitions into more than two vertex classes. Our proofs yield polynomial algorithms.

Abstract: We show that a random graph studied by Ioffe and Levit is an example of an inhomogeneous random graph of the type studied by Bollobas, Janson and Riordan, which enables us to give a new, and perhaps more revealing, proof of their result on a phase transition

Abstract: We say that $n$-vertex graphs $G_1,G_2,\ldots,G_k$ pack if there exist injective mappings of their vertex sets onto $[n] = \{1, \ldots,n \}$ such that the images of the edge sets do not intersect. The notion of packing allows one to make some problems on graphs more natural or more general. Clearly, two $n$-vertex graphs $G_1$ and $G_2$ pack if and only if $G_1$ is a subgraph of the complement $\overline{G}_2$ of $G_2$.

Abstract: A widely studied model for generating binary sequences is to ‘evolve’ them on a tree according to a symmetric Markov process. We show that under this model distinguishing the true (model) tree from a false one is substantially ‘easier’ (in terms of the sequence length needed) than determining the true tree. The key tool is a new and near-tight Ramsey-type result for binary trees.

Abstract: Let d =1≤ d 1 ≤ d 2 ≤···.≤ d n be a non-decreasing sequence of n positive integers, whose sum is even. Let denote the set of graphs with vertex set [ n ]={1,2,. . .., n } in which the degree of vertex i is d i . Let G n , d be chosen uniformly at random from . Let d =( d 1 + d 2 +···.+ d n )/ n be the average degree. We give a condition on d under which we can show that w.h.p. the chromatic number of is Θ( d /ln d ). This condition is satisfied by graphs with exponential tails as well those with power law tails.

Abstract: 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 $K_2\,{\square}\, K_n$ is determined.

Abstract: 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 Kagome bond model we improve bounds from $0.5182 \leq p_c \leq 0.5335$ to $0.522197 \leq p_c \leq 0.526873$, improving the range from 0.0153 to 0.004676. For the $(3,12^2)$ bond model we improve bounds from $0.7393 \leq p_c \leq 0.7418$ to $0.739773 \leq p_c \leq 0.741125$, 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.

Abstract: 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.

Abstract: We study semirandom k-colourable graphs made up as follows. Partition the vertex set V = {1, . . ., n} randomly into k classes V1, . . ., Vk of equal size and include each Vi–Vj-edge with probability p independently (1 ≤ i )klnn, C0k2} for a certain constant C0>0 and an arbitrarily small but constant >0, an optimal colouring of G*n,p,k can be found in polynomial time with high probability. Furthermore, if np ≥ C0max{klnn, k2}, a k-colouring of G*n,p,k can be computed in polynomial expected time. Moreover, an optimal colouring of G*n,p,k can be computed in expected polynomial time if k ≤ ln1/3n and np ≥ C0klnn. By contrast, it is NP-hard to k-colour G*n,p,k With high probability if $np\leq (\frac12-\varepsilon)k\ln(n/k)$ .

Abstract: The chromatic polynomial PΓ(x) of a graph “ is a polynomial whose value at the positive integer k is the number of proper k-colourings of Γ. If G is a group of automorphisms of Γ, then there is a polynomial OPΓ,G(x), whose value at the positive integer k is the number of orbits of G on proper k-colourings of “. It is known that real chromatic roots cannot be negative, but they are dense in [32/27·∞). Here we discuss the location of real orbital chromatic roots. We show, for example, that they are dense in ℝ, but under certain hypotheses, there are zero-free regions. We also look at orbital flow roots. Here things are more complicated because the orbit count is given by a multivariate polynomial; but it has a natural univariate specialization, and we show that the roots of these polynomials are dense in the negative real axis.

Abstract: Let $P(G,t)$ and $F(G,t)$ denote the chromatic and flow polynomials of a graph $G$. G. D. Birkhoff and D C. Lewis showed that, if $G$ is a plane near-triangulation, then the only zeros of $P(G,t)$ in $(-\infty,2]$ are 0, 1 and 2. We will extend their theorem by showing that a stronger result to the dual statement holds for both planar and non-planar graphs: if $G$ is a bridge graph with at most one vertex of degree other than three, then the only zeros of $F(G,t)$ in $(-\infty,\alpha]$ are 1 and 2, where $\alpha\approx 2.225\cdots$ is the real zero in $(2,3)$ of the polynomial $t^4-8t^3+22t^2-28t+17$. In addition we construct a sequence of ‘near-cubic’ graphs whose flow polynomials have zeros converging to $\alpha$ from above.

Abstract: This paper is devoted to an online variant of the minimum spanning tree problem in randomly weighted graphs. We assume that the input graph is complete and the edge weights are uniformly distributed over [0,1]. An algorithm receives the edges one by one and has to decide immediately whether to include the current edge into the spanning tree or to reject it. The corresponding edge sequence is determined by some adversary. We propose an algorithm which achieves $\mathbb{E}[ALG]/\mathbb{E}[OPT]=O(1)$ and $\mathbb{E}[ALG/OPT]=O(1)$ against a fair adaptive adversary, i.e., an adversary which determines the edge order online and is fair in a sense that he does not know more about the edge weights than the algorithm. Furthermore, we prove that no online algorithm performs better than $\mathbb{E}[ALG]/\mathbb{E}[OPT]=\Omega(\log n)$ if the adversary knows the edge weights in advance. This lower bound is tight, since there is an algorithm which yields $\mathbb{E}[ALG]/\mathbb{E}[OPT]=O(\log n)$ against the strongest-imaginable adversary.

Abstract: Consider the set of finite words on a totally ordered alphabet with two letters. We prove that the distribution of the length of the standard right factor of a random Lyndon word with length n, divided by n, converges to u(dx) = ½δ1(dx + ½1(0,1)(x)dx when n goes to infinity. The convergence of all moments follows. This paper thus completes the results of [2], in which the limit of the first moment is given.

Abstract: Let D(G) be the smallest quantifier depth of a first-order formula which is true for a graph G but false for any other non-isomorphic graph. This can be viewed as a measure for the descriptive complexity of G in first-order logic. We show that almost surely D(G)=θ(lnn / ln ln n), where G is a random tree of order n or the giant component of a random graph G(n,c/n) with constant c<1. These results rely on computing the maximum of D(T) for a tree T of order n and maximum degree l, so we study this problem as well.