The lamplighter group over $\Z$ is the wreath product $\Z_q \wr \Z$. With respect to a natural generating set, its Cayley graph is the Diestel–Leader graph $\mbox{\sl DL}(q,q)$. We study harmonic functions for the ‘simple’ Laplacian on this graph and, more generally, for a class of random walks on $\mbox{\sl DL}(q,r)$, where $q,r \geq 2$. The $\mbox{\sl DL}$-graphs are horocyclic products of two trees, and we give a full description of all positive harmonic functions in terms of the boundaries of these two trees. In particular, we determine the minimal Martin boundary, that is, the set of minimal positive harmonic functions.

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.

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.

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.

In 1978 Erdős, Faudree, Rousseau and Schelp conjectured that \[ r ( C_{p},K_{r} ) = ( p-1 ) (r-1) +1 \] for every $p\,{\geq}\,r\,{\geq}\,3$, except for $p\,{=}\,q\,{=}\,3$. This has been proved for $r\,{\leq}\,6$, and for \[ p \geq r^{2}-2r\].

In this note we prove the conjecture for $p\,{\geq}\,4r+2$.

We prove that for all $\alpha,c>0$ and for all bipartite graphs $H$, all but at most $\alpha n$ vertices of every $cn$-regular graph $G$ whose order $n$ is sufficiently large can be covered by vertex-disjoint copies of $H$. If the vertex classes of $H$ have different size, then even all but a constant number of vertices of $G$ can be covered. This implies that for all $c>0$ and all $r\geq 4$ there exists a constant $C$ such that, in every $cn$-regular graph $G$, all but at most $C$ vertices can be covered by vertex-disjoint subdivisions of $K_r$. We also show that for $r=4,5$ one can take $C=0$.

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 $g \in L^{-1}(i)$, $$ p_{ij} = \sum_{h \in L^{-1}(j)} \!\! \mu (g^{-1}h).$$

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{$k:Q(n,k)=\theta(\frac{n^2}{k})$}.

Randomized search heuristics like evolutionary algorithms and simulated annealing find many applications, especially in situations where no full information on the problem instance is available. In order to understand how these heuristics work, it is necessary to analyse their behaviour on classes of functions. Such an analysis is performed here for the class of monotone pseudo-Boolean polynomials. Results depending on the degree and the number of terms of the polynomial are obtained. The class of monotone polynomials is of special interest since simple functions of this kind can have an image set of exponential size, improvements can increase the Hamming distance to the optimum and, in order to find a better search point, it can be necessary to search within a large plateau of search points with the same fitness value.

Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new ‘singular’ approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of $S^1$ in the space formed by the graph together with its ends.

Our approach permits the extension to infinite graphs of standard results about finite graph homology – such as cycle–cocycle duality and Whitney's theorem, Tutte's generating theorem, MacLane's planarity criterion, the Tutte/Nash-Williams tree packing theorem – whose infinite versions would otherwise fail. A notion of end degrees motivated by these results opens up new possibilities for an ‘extremal’ branch of infinite graph theory.

Numerous open problems are suggested.

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.

This paper is an extended version of the lecture given by the second author delivered at the conference ‘Combinatorics: Walter Deuber Memorial Meeting’ held on 7–8 October 2002 at Humboldt-Universität zu Berlin. Regretfully, this topic was to be the last that fascinated Walter Deuber's mind. We wrote this article in remembrance of that.

Let ${\cal T}(n,m)$ denote the set of all labelled triangle-free graphs with $n$ vertices and exactly $m$ edges. In this paper we give a short self-contained proof of the fact that there exists a constant $C>0$ such that, for all $m\geq Cn^{3/2}\sqrt{\log n}$, a graph chosen uniformly at random from ${\cal T}(n,m)$ is with probability $1-o(1)$ bipartite.

We consider several extremal problems concerning representations of graphs as distance graphs on the integers. Given a graph $G=(V,E)$, we wish to find an injective function $\phi:V\to{\mathbb Z}^+=\{1,2,\dots\}$ and a set ${\mathcal D}\subset{\mathbb Z}^+$ such that $\{u,v\}\in E$ if and only if $|\phi(u)-\phi(v)|\in{\mathcal D}$.

Let $s(n)$ be the smallest $N$ such that any graph $G$ on $n$ vertices admits a representation $(\phi_G,{\mathcal D}_G)$ such that $\phi_G(v)\leq N$ for all $v\in V(G)$. We show that $s(n)=(1+o(1))n^2$ as $n\to\infty$. In fact, if we let $s_r(n)$ be the smallest $N$ such that any $r$-regular graph $G$ on $n$ vertices admits a representation $(\phi_G,{\mathcal D}_G)$ such that $\phi_G(v)\leq N$ for all $v\in V(G)$, then $s_r(n)=(1+o(1))n^2$ as $n\to\infty$ for any $r=r(n)\gg\log n$ with $rn$ even for all $n$.

Given a graph $G=(V,E)$, let $D_{\rm e}(G)$ be the smallest possible cardinality of a set ${\mathcal D}$ for which there is some $\phi\:V\to{\mathbb Z}^+$ so that $(\phi,{\mathcal D})$ represents $G$. We show that, for almost all $n$-vertex graphs $G$, we have \begin{equation*} D_{\rm e}(G)\geq\frac{1}{2}\binom{n}{2}-(1+o(1))n^{3/2}(\log n)^{1/2}, \end{equation*} whereas for some $n$-vertex graph $G$, we have \begin{equation*} D_{\rm e}(G)\geq\binom{n}{2}-n^{3/2}(\log n)^{1/2+o(1)}.\end{equation*} Further extremal problems of similar nature are considered.

We present a programme of characterizing Ramsey classes of structures by a combination of the model theory and combinatorics. In particular, we relate the classification programme of countable homogeneous structures (of Lachlan and Cherlin) to the classification of Ramsey classes. As particular instances of this approach we characterize all Ramsey classes of graphs, tournaments and partial ordered sets. We fully characterize all monotone Ramsey classes of relational systems (of any type). We also carefully discuss the role of (admissible) orderings which lead to a new classification of Ramsey properties by means of classes of order-invariant objects.

The aim of this paper is to point to a difference between binary and hyperary structures. The modular counting functions of a class of structures defined by a sentence of second-order monadic logic with equality, based on binary relations, are ultimately periodic. However, this is not the case for sentences based on quaternary relations.

Let $G$ be a noncomplete $k$-connected graph such that the graphs obtained from contracting any edge in $G$ are not $k$-connected, and let $t(G)$ denote the number of triangles in $G$. Thomassen proved $t(G) \geq 1$, which was later improved by Mader to $t(G) \geq \frac{1}{3}|V(G)|$.

Here we show $t(G) \geq \frac{2}{3}|V(G)|$ (which is best possible in general).

Furthermore it is proved that, for $k \geq 4$, a $k$-connected graph without two disjoint triangles must contain an edge not contained in a triangle whose contraction yields a $k$-connected graph. As an application, for $k \geq 4$ every $k$-connected graph $G$ admits two disjoint induced cycles $C_1,C_2$ such that $G-V(C_1)$ and $G-V(C_2)$ are $(k-3)$-connected.

We consider algorithms for group testing problems when nothing is known in advance about the number of defectives. Du and Hwang suggested measuring the quality of such algorithms by its so-called (first) competitive ratio (see the Introduction). Later, Du and Park suggested a second kind of competitive ratio. For each kind of competitiveness, we improve the best-known bounds: in the first case, from 1.65 to $1.5+\ep$, and in the second from 16 to 4.

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 $RS(3,t)\geq t^2-3$.

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.