We study the Lovász number $\vartheta$ along with two related SDP relaxations $\vartheta_{1/2}$, $\vartheta_2$ of the independence number and the corresponding relaxations $\bar\vartheta$, $\bar\vartheta_{1/2}$, $\bar\vartheta_2$ of the chromatic number on random graphs $G_{n,p}$. We prove that $\vartheta,\vartheta_{1/2},\vartheta_2(G_{n,p})$ are concentrated about their means, and that $\bar\vartheta,\bar\vartheta_{1/2},\bar\vartheta_2(G_{n,p})$ in the case $p<n^{-1/2-\varepsilon}$ are concentrated in intervals of constant length. Moreover, extending a result of Juhász [28], we estimate the probable value of $\vartheta,\vartheta_{1/2},\vartheta_2(G_{n,p})$ for edge probabilities $c_0/n\leq p\leq 1-c_0/n$, where $c_0>0$ is a constant. As an application, we give improved algorithms for approximating the independence number of $G_{n,p}$ and for deciding $k$-colourability in polynomial expected time.

For a graph G, let f(G) denote the maximum number of edges in a cut of G. For an integer m and for a fixed graph H, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$, as G ranges over all graphs on m edges that contain no copy of H. In this paper we study this function for various graphs H. In particular we show that for any graph H obtained by connecting a single vertex to all vertices of a fixed nontrivial forest, there is a $c(H) >0$ such that $f(m,H) \geq \frac{m}{2} + c(H) m^{4/5}$, and that this is tight up to the value of $c(H)$. We also prove that for any even cycle $C_{2k}$ there is a $c(k)>0$ such that $f(m,C_{2k}) \geq \frac{m}{2} + c(k) m^{(2k+1)/(2k+2)}$, and that this is tight, up to the value of $c(k)$, for $2k\in \{4,6,10\}$. The proofs combine combinatorial, probabilistic and spectral techniques.

Let $T$ denote a real function defined on random subsets of a given family of finite sets. The random variable $T$ is decomposed into the sum of the linear, the quadratic, the cubic etc. parts which are mutually uncorrelated. Applications of this decomposition to the asymptotics of the probability distribution of $T$ (as the sizes of random subsets and of finite sets increase) are discussed.

Walter Deuber died on 16th July 1999 at the age of 56 after a one-and-a-half year struggle with cancer. On 6th October 2002 he would have celebrated his 60th birthday. In order to commemorate this date several of his friends, former students and colleagues came together in the evening of this day in Berlin and started a two-day conference on Combinatorics in honour of Walter Deuber.

Complete disorder is impossible – this theme of Ramsey Theory, as stated by Theodore S. Motzkin, was a guiding theme throughout Walter Deuber's scientific life.

In a 1947 paper [6], M. Kac derived the eigenvalues and eigenvectors of the probability transition matrix associated with the Ehrenfest urn model with $n$ balls, in which a ball is selected at random and moved to the other urn. The connection (see, for instance, [1, p. 19]) between the Markov chain defined by the Ehrenfest model and the nearest-neighbour uniform random walk on the abelian group $\Z^n_2$ prompted M. Kac to ask when a Markov chain may be lifted to a random walk on a group. More specifically, given a Markov chain $\{X_m\}_{m \geq 0}$ with state space $S = \{1, 2, \ldots, n \}$ and probability transition matrix ${\mbox{\bf $P$}} = [p_{ij}] (\mbox{here } p_{ij}:= P(X_{r + 1} = j | X_r = i) \mbox{ for all } r \geq 0)$, we say that the Markov chain $\{X_m\}_{m \geq 0}$ or equivalently ${\mbox{\bf $P$}}$lifts to a random walk on a finite group $G$ if there exists a probability measure $\mu$ on $G$ and a surjective map $L:G \rightarrow S$ such that, for all $i, j \in S$, and for each $g \in L^{-1}(i)$, $$ p_{ij} = \sum_{h \in L^{-1}(j)} \!\! \mu (g^{-1}h).$$

This paper is partly an overview of the subject (Sections 1–4), and partly a research paper containing proofs for new results (Sections 5–7).

A Fano configuration is the hypergraph of 7 vertices and 7 triplets defined by the points and lines of the finite projective plane of order 2. Proving a conjecture of T. Sós, the largest triple system on $n$ vertices containing no Fano configuration is determined (for $n> n_1$). It is 2-chromatic with $\binom{n}{3}-\binom{\lfloor n/2 \rfloor}{3} -\binom{\lceil n/2 \rceil}{3}$ triples. This is one of the very few nontrivial exact results for hypergraph extremal problems.

Suppose we are given $n$ coloured balls and an integer $k$ between 2 and $n$. How many colour-comparisons $Q(n,k)$ are needed to decide whether $k$ balls have the same colour? The corresponding problem when there is an (unknown) linear order with repetitions on the balls was solved asymptotically by Björner, Lovász and Yao, the complexity being \smash{$\theta (n\log\frac{2n}{k})$}. Here we give the exact answer for \smash{$k>\frac{n}{2}: Q(n,k)=2n-k-1$}, and the order of magnitude for arbitrary \smash{$k:Q(n,k)=\theta(\frac{n^2}{k})$}.

A detachment of a graph $G$ is formed by splitting each vertex into one or more subvertices, and sharing the incident edges arbitrarily among the subvertices. In this paper we consider the question of whether a graph $H$ is a detachment of some complete graph $K_n$. When $H$ is large and restricted to belong to certain classes of graphs, for example bounded degree planar triangle-free graphs, we obtain necessary and sufficient conditions which give a complete characterization.

A harmonious colouring of a simple graph $G$ is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number$h(G)$ is the least number of colours in such a colouring. The results on detachments of complete graphs give exact results on harmonious chromatic number for many classes of graphs, as well as algorithmic results.

We introduce a family of one-dimensional geometric growth models, constructed iteratively by locally optimizing the trade-offs between two competing metrics, and show that this family is equivalent to a family of preferential attachment random graph models with upper cut-offs. This is the first explanation of how preferential attachment can arise from a more basic underlying mechanism of local competition. We rigorously determine the degree distribution for the family of random graph models, showing that it obeys a power law up to a finite threshold and decays exponentially above this threshold.

We also rigorously analyse a generalized version of our graph process, with two natural parameters, one corresponding to the cut-off and the other a ‘fertility’ parameter. We prove that the general model has a power-law degree distribution up to a cut-off, and establish monotonicity of the power as a function of the two parameters. Limiting cases of the general model include the standard preferential attachment model without cut-off and the uniform attachment model.

Let $G$ be a finite group of order $n$ and let $k$ be a natural number. Let $\{x_i : i\in I\}$ be a family of elements of $G$ such that $|I|= n+k-1$. Let $v$ be the most repeated value of the family. Let $ \{ \sigma_i : 1\leq i \leq k \} $ be a family of permutations of $G$ such that $\sigma_i(1)=1$ for all $i$. We obtain the following result.

There are pairwise distinct elements $i_1, i_2, \dots ,i_k\in I$ such that \[ \prod_{1\leq j\leq k } \sigma_j \big(v^{-1}x_ {i_j }\big) =1.\]

The Ramsey Schur number $RS(s,t)$ is the smallest $n$ such that every 2-colouring of the edges of $K_n$ with vertices $1,2,\ldots,n$ contains a green $K_s$ or there are vertices $x_1,x_2,\ldots,x_t$ fulfilling the equation $x_1+x_2+\cdots+x_{t-1}=x_t$ and all edges $(x_i,x_j)$ are red. We prove $RS(3,3)=11, RS(3,t)=t^2-3$ for $t\equiv1\ (\mbox{mod}\ 6)$ and $t=8$, and $RS(3,t)\geq t^2-3$.

We give new formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay [Europ. J. Combin. 4 149–160] for regular graphs. The general answer involves a quantity for infinite graphs that we call ‘tree entropy’, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasi-transitive amenable graphs, extending a result of Burton and Pemantle [Ann. Probab. 21 1329–1371].

In 1978, Bollobás and Eldridge [5] made the following two conjectures.

1. (C1) There exists an absolute constant $c>0$ such that, if k is a positive integer and $G_1$ and $G_2$ are graphs of order n such that $\Delta(G_1),\Delta(G_2)\leq n-k$ and $e(G_1),e(G_2)\leq ck n$, then the graphs $G_1$ and $G_2$ pack.

2. (C2) For all $0<\alpha<1/2$ and $0<c<\sqrt{1/8}$, there exists an $n_0=n_0(\alpha,c)$ such that, if $G_1$ and $G_2$ are graphs of order $n>n_0$ satisfying $e(G_1)\leq \alpha n$ and $e(G_2)\leq c\sqrt{n^3/ \alpha}$, then the graphs $G_1$ and $G_2$ pack.

A stable set $I$ of a graph $G$ is called $k$-extendable, $k \,{\ge}\, 1$, if there exists a stable set $X \,{\subseteq}\,V(G) {\setminus} I$ such that $|X| \,{\le}\, k$ and $|N(X) \,{\cap}\, I| \,{<}\, |X|$. A graph $G$ is called $k$-extendable if every stable set in $G$, which is not maximum, is $k$-extendable. Let us denote by ${\rm E}(k)$ the class of all $k$-extendable graphs.

We present a finite forbidden induced subgraph characterization of the maximal hereditary subclass ${\rm PE}(k)$ in ${\rm E}(k)$ for every $k \,{\ge}\,1$.

Thus, we define a hierarchy ${\rm PE}(1) \,{\subset}\, {\rm PE}(2) \,{\subset}\,{\cdots}\,{ \subset}\, {\rm PE}(k) \,{\subset}\,{ \cdots}\,$ of hereditary classes of graphs, in each of which a maximum stable set can be found in polynomial time. The hierarchy covers all graphs, and all its classes can be recognized in polynomial time.

Motivated by the result that an ‘approximate’ evaluation of the Jones polynomial of a braid at a 5th root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP to the counting class GapP, we introduce a form of additive approximation which can be used to simulate a function in BQP. We show that all functions in the classes #P and GapP have such an approximation scheme under certain natural normalizations. However, we are unable to determine whether the particular functions we are motivated by, such as the above evaluation of the Jones polynomial, can be approximated in this way. We close with some open problems motivated by this work.

We derive a generalization of a theorem of Raimi proving there is a partition of natural numbers with given densities of classes which meet structured translates of any other class of a partition of natural numbers.

A zigzag in a plane graph is a circuit of edges, such that any two, but not three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbours on opposite edges. A graph without a railroad is called tight. We consider the zigzag and railroad structures of general 3-valent plane graph and, especially, of simple two-faced polyhedra, i.e., 3-valent 3-polytopes with only $a$-gonal and $b$-gonal faces, where $3 \leq a < b \leq 6$; the main cases are $(a,b)=(3,6), (4,6)$ and $(5,6)$ (the fullerenes).

We completely describe the zigzag structure for the case $(a,b)\,{=}\,(3,6)$. For the case $(a,b)\,{=}\,(4,6)$ we describe symmetry groups, classify all tight graphs with simple zigzags and give the upper bound 9 for the number of zigzags in general tight graphs. For the remaining case $(a,b)\,{=}\,(5,6)$ we give a construction realizing a prescribed zigzag structure.

We study self-avoiding walks (SAWs) on non-Euclidean lattices that correspond to regular tilings of the hyperbolic plane (‘hyperbolic graphs’). We prove that on all but at most eight such graphs, (i) there are exponentially fewer $N$-step self-avoiding polygons than there are $N$-step SAWs, (ii) the number of $N$-step SAWs grows as $\mu_w^N$ within a constant factor, and (iii) the average end-to-end distance of an $N$-step SAW is approximately proportional to $N$. In terms of critical exponents from statistical physics, (ii) says that $\gamma=1$ and (iii) says that $\nu=1$. We also prove that $\gamma$ is finite on all hyperbolic graphs, and we prove a general identity about non-reversing walks that had previously been discovered for certain special cases.