We present versions of concentration inequalities for products of Markov kernels and graph products. We also present discussions of a variety of consequences such as sharp upper bounds, in terms of the diameter of the state space, on the spectral gap.

Simulated annealing is a very successful heuristic for various problems in combinatorial optimization. In this paper an application of simulated annealing to the 3-colouring problem is considered. In contrast to many good empirical results we will show for a certain class of graphs that the expected first hitting time of a proper colouring, given an arbitrary cooling scheme, is of exponential size.

These results are complementary to those in [13], where we prove the convergence of simulated annealing to an optimal solution in exponential time.

Fix a small graph H and let YH denote the number of copies of H in the random graph G(n, p). We investigate the degree of concentration of YH around its mean, motivated by the following questions.

[bull] What is the upper tail probability Pr(YH [ges ] (1 + ε)[](YH))?

[bull] For which λ does YH have sub-Gaussian behaviour, namely

(formula here)

where c is a positive constant?

[bull] Fixing λ = ω(1) in advance, find a reasonably small tail T = T(λ) such that

(formula here)

We prove a general concentration result which contains a partial answer to each of these questions. The heart of the proof is a new martingale inequality, due to J. H. Kim and the present author [13].

We show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree [les ] r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc [mid ]q[mid ] < C(r). Furthermore, C(r) [les ] 7.963907r. This result is a corollary of a more general result on the zeros of the Potts-model partition function ZG(q, {ve}) in the complex antiferromagnetic regime [mid ]1 + ve[mid ] [les ] 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Kotecký–Preiss condition for nonvanishing of a polymer-model partition function. We also show that, for all loopless graphs G of second-largest degree [les ] r, the zeros of PG(q) lie in the disc [mid ]q[mid ] < C(r) + 1. Along the way, I give a simple proof of a generalized (multivariate) Brown–Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.

This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth, Sum Cut, Vertex Separation, Edge Bisection and Vertex Bisection. For full square lattices, we give optimal layouts for the problems still open. For arbitrary lattice graphs, we present best possible bounds disregarding a constant factor. We apply percolation theory to the study of lattice graphs in a probabilistic setting. In particular, we deal with the subcritical regime that this class of graphs exhibits and characterize the behaviour of several layout measures in this space of probability. We extend the results on random lattice graphs to random geometric graphs, which are graphs whose nodes are spread at random in the unit square and whose edges connect pairs of points which are within a given distance. We also characterize the behaviour of several layout measures on random geometric graphs in their subcritical regime. Our main results are convergence theorems that can be viewed as an analogue of the Beardwood, Halton and Hammersley theorem for the Euclidean TSP on random points in the unit square.

Let G be an abelian group. For a subset A ⊂ G, denote by 2 ∧ A the set of sums of two different elements of A. A conjecture by Erdős and Heilbronn, first proved by Dias da Silva and Hamidoune, states that, when G has prime order, [mid ]2 ∧ A[mid ] [ges ] min([mid ]G[mid ], 2[mid ]A[mid ] − 3).

We prove that, for abelian groups of odd order (respectively, cyclic groups), the inequality [mid ]2 ∧ A[mid ] [ges ] min([mid ]G[mid ], 3[mid ]A[mid ]/2) holds when A is a generating set of G, 0 ∈ A and [mid ]A[mid ] [ges ] 21 (respectively, [mid ]A[mid ] [ges ] 33). The structure of the sets for which equality holds is also determined.

A graph G is quasi 4-connected if it is simple, 3-connected, has at least five vertices, and for every partition (A, B, C) of V(G) either [mid ]C[mid ] [ges ] 4, or G has an edge with one end in A and the other end in B, or one of A,B has at most one vertex. We show that any quasi 4-connected nonplanar graph with minimum degree at least three and no cycle of length less than five has a minor isomorphic to P−10, the Petersen graph with one edge deleted. We deduce the following weakening of Tutte's Four Flow Conjecture: every 2-edge-connected graph with no minor isomorphic to P−10 has a nowhere-zero 4-flow. This extends a result of Kilakos and Shepherd who proved the same for 3-regular graphs.

Consider sequences {Xi}mi=1 and {Yj}nj=1 of independent random variables, taking values in a finite alphabet, and assume that the variables X1, X2, … and Y1, Y2, … follow the distributions μ and v, respectively. Two variables Xi and Yj are said to match if Xi = Yj. Let the number of matching subsequences of length k between the two sequences, when r, 0 [les ] r < k, mismatches are allowed, be denoted by W.

In this paper we use Stein's method to bound the total variation distance between the distribution of W and a suitably chosen compound Poisson distribution. To derive rates of convergence, the case where E[W] stays bounded away from infinity, and the case where E[W] → ∞ as m, n → ∞, have to be treated separately. Under the assumption that ln n/ln(mn) → ρ ∈ (0, 1), we give conditions on the rate at which k → ∞, and on the distributions μ and v, for which the variation distance tends to zero.

Let 0 < p < 1, q = 1 − p and b be fixed and let G ∈ [Gscr](n, p) be a graph on n vertices where each pair of vertices is joined independently with probability p. We show that the probability that every vertex of G has degree at most pn + b √npq is equal to (c + o(1))n, for c = c(b) the root of a certain equation. Surprisingly, c(0) = 0.6102 … is greater than ½ and c(b) is independent of p. To obtain these results we consider the complete graph on n vertices with weights on the edges. Taking these weights as independent normal N(p, pq) random variables gives a ‘continuous’ approximation to [Gscr](n, p) whose degrees are much easier to analyse.

An asymptotics, as n → ∞, for the expected number of distinct part sizes in a random composition of an integer n is obtained.

A family of subsets of an n-set is k-locally thin if, for every k of its member sets, the ground set has at least one element contained in exactly 1 of them. We derive new asymptotic upper bounds for the maximum cardinality of locally thin set families for every even k. This improves on previous results of two of the authors with Monti.