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 $C=0$.

If all nonzero eigenvalues of the (normalized) Laplacian of a graph $G$ are close to 1, then $G$ is $t$-Turán in the sense that any subgraph of $G$ containing no $K_{t+1}$ contains at most $(1-1/t + o(1) ) e(G)$ edges where $e(G)$ denotes the number of edges in G.

We show that symmetry, represented by a graph's automorphism group, can be used to greatly reduce the computational work for the substitution method. This allows application of the substitution method over larger regions of the problem lattices, resulting in tighter bounds on the percolation threshold $p_c$. We demonstrate the symmetry reduction technique using bond percolation on the $(3,12^2)$ lattice, where we improve the bounds on $p_c$ from (0.738598,0.744900) to (0.739399,0.741757), a reduction of more than 62% in width, from 0.006302 to 0.002358.

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.

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.

In a one-parameter model for evolution of random trees, which also includes the Barabási–Albert random graph [1], the law of large numbers and the central limit theorem are proved for the maximal degree. In the proofs martingale methods are applied.

Suppose that $q$ is a prime power exceeding five. For every integer $N$ there exists a 3-connected GF($q$)-representable matroid that has at least $N$ inequivalent GF($q$)-representations. In contrast to this, Geelen, Oxley, Vertigan and Whittle have conjectured that, for any integer $r > 2$, there exists an integer $n(q,\, r)$ such that if $M$ is a 3-connected GF($q$)-representable matroid and $M$ has no rank-$r$ free-swirl or rank-$r$ free-spike minor, then $M$ has at most $n(q,\, r)$ inequivalent GF($q$)-representations. The main result of this paper is a proof of this conjecture for Zaslavsky's class of bias matroids.

Let $P(n)$ and $C(n)$ denote, respectively, the maximum possible numbers of Hamiltonian paths and Hamiltonian cycles in a tournament on n vertices. The study of $P(n)$ was suggested by Szele [14], who showed in an early application of the probabilistic method that $P(n) \geq n!2^{-n+1}$, and conjectured that $\lim ( {P(n)}/ {n!} )^{1/n}= 1/2.$ This was proved by Alon [2], who observed that the conjecture follows from a suitable bound on $C(n)$, and showed $C(n) <O(n^{3/2}(n-1)!2^{-n}).$ Here we improve this to $C(n)<O\big(n^{3/2-\xi}(n-1)!2^{-n}\big),$ with $\xi = 0.2507$… Our approach is mainly based on entropy considerations.

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 $p\,{\geq}\,4r+2$.

JumbleG is a Maker–Breaker game. Maker and Breaker take turns in choosing edges from the complete graph $K_n$. Maker's aim is to choose what we call an $\epsilon$-regular graph (that is, the minimum degree is at least $(\frac12-\epsilon) n$ and, for every pair of disjoint subsets $S,T\subset V$ of cardinalities at least $\epsilon n$, the number of edges $e(S,T)$ between $S$ and $T$ satisfies $\bigl|\frac{e(S,T)}{|S|\,|T|}-\frac12\bigr|\leq \epsilon$.) In this paper we show that Maker can create an $\epsilon$-regular graph, for $\epsilon\geq 2(\log n/n)^{1/3}$. We also consider a similar game, JumbleG2, where Maker's aim is to create a graph with minimum degree at least $\bigl(\frac12-\epsilon\bigr)n$ and maximum co-degree at most $\bigl(\frac14+\epsilon\bigr)n$, and show that Maker has a winning strategy for $\epsilon> 3 (\log n/n)^{1/2}$. Thus, in both games Maker can create a pseudo-random graph of density $\frac12$. This guarantees Maker's win in several other positional games, also discussed here.

The present paper deals with two graph parameters related to cover graphs and acyclic orientations of graphs.

The parameter $c(G)$ of a graph $G$, introduced by B. Bollobás, G. Brightwell and J. Nešetřil [Order3 245–255], is defined as the minimum number of edges one needs to delete from $G$ in order to obtain a cover graph. Extending their results, we prove that, for $\delta >0$, $(1-\delta) \frac{1}{l} \frac{n^2p}{2} \leq c({\mathcal G}_{n,p}) \leq (1+\delta) \frac{1}{l} \frac{n^2p}{2}$ asymptotically almost surely as long as $C n^{-1 + \frac{1}{l}} \leq p(n) \leq c n^{-1 + \frac{1}{ l-1} }$ for some positive constants $c$ and $C$. Here, as usual, ${\mathcal G}_{n,p}$ is the random graph.

Given an acyclic orientation of a graph $G$, an arc is called dependent if its reversal creates an oriented cycle. Let $d_{\min}(G)$ be the minimum number of dependent arcs in any acyclic orientation of $G$. We determine the supremum, denoted by $r_{\chi,g}$, of $d_{\min}(G)/e(G)$ in the class of graphs $G$ with chromatic number $\chi$ and girth $g$. Namely, we show that $r_{\chi,g} = {(\scriptsize\begin{array}{@{}c@{}}{\chi}-g+2\\ 2\end{array})} / {(\scriptsize\begin{array}{@{}c@{}}{\chi}\\ 2\end{array})}$. This extends results of D. C. Fisher, K. Fraughnaugh, L. Langley and D. B. West [J. Combin. Theory Ser. B71 73–78].

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.

Many applications of Szemerédi's Regularity Lemma for graphs are based on the following counting result. If ${\mathcal G}$ is an $s$-partite graph with partition $V({\mathcal G}) =\bigcup_{i=1}^{s} V_i$, $\vert V_i\vert =m$ for all $i\in [s]$, and all pairs $(V_i, V_j)$, $1\leq i < j\leq s$, are $\epsilon$-regular of density $d$, then $\mathcal{G}$ contains $(1\pm f(\epsilon))d^{({s\atop 2})}m^s$ cliques $K_{s}$, provided $\epsilon<\epsilon(d)$, where $f(\epsilon)$ tends to 0 as $\epsilon$ tends to 0.

Guided by the regularity lemma for 3-uniform hypergraphs established earlier by Frankl and Rödl, Nagle and Rödl proved a corresponding counting lemma. Their proof is rather technical, mostly due to the fact that the ‘quasi-random’ hypergraph arising after application of Frankl and Rödl's regularity lemma is ‘sparse’, and consequently difficult to handle.

When the ‘quasi-random’ hypergraph is ‘dense’ Kohayakawa, Rödl and Skokan (J. Combin. Theory Ser. A97 307–352) found a simpler proof of the counting lemma. Their result applies even to $k$-uniform hypergraphs for arbitrary $k$. While the Frankl–Rödl regularity lemma will not render the dense case, in this paper, for $k=3$, we are nevertheless able to reduce the harder, sparse case to the dense case.

Namely, we prove that a ‘dense substructure’ randomly chosen from the ‘sparse $\delta$-regular structure’ is $\delta$-regular as well. This allows us to count the number of cliques (and other subhypergraphs) using the Kohayakawa–Rödl–Skokan result, and provides an alternative proof of the counting lemma in the sparse case. Since the counting lemma in the dense case applies to $k$-uniform hypergraphs for arbitrary $k$, there is a possibility that the approach of this paper can be adopted to the general case as well.

Let ${\cal H}$ be a 3-uniform hypergraph on an $n$-element vertex set $V$. The neighbourhood of $a,b\in V$ is $N(ab):= \{x: abx\in E({\cal H})\} $. Such a 3-graph has independent neighbourhoods if no $N(ab)$ contains an edge of ${\cal H}$. This is equivalent to ${\cal H}$ not containing a copy of $\mathbb{F} :=\{ abx$, $aby$, $abz$, $xyz\}$.

In this paper we prove an analogue of the Andrásfai–Erdös–Sós theorem for triangle-free graphs with minimum degree exceeding $2n/5$. It is shown that any $\mathbb{F}$-free 3-graph with minimum degree exceeding $(\frac{4}{9}-\frac{1}{125})\binom{n}{2}$ is bipartite, (for $n> n_0$), i.e., the vertices of ${\cal H}$ can be split into two parts so that every triple meets both parts.

This is, in fact, a Turán-type result. It solves a problem of Erdös and T.Sós, and answers a question of Mubayi and Rödl that

\[\ex(n,\mathbb{F}_{3,2})=\max\limits_{\alpha}(n-\alpha)\left({\alpha\atop2}\right)\].

Here the right-hand side is $\frac{4}{9}\binom{n}{3}+O(n^2)$. Moreover $e({\cal H})={\rm ex}(n,\mathbb{F})$ is possible only if $V({\cal H})$ can be partitioned into two sets $A$ and $B$ so that each triple of ${\cal H}$ intersects $A$ in exactly two vertices and $B$ in one.

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 extend a result by Füredi and Komlós and show that the first eigenvalue of a random graph is asymptotically normal, both for $G_{n,p}$ and $G_{n,m}$, provided $np\geq n^\delta$ or $m/n\geq n^\delta$ for some $\delta>0$. The asymptotic variance is of order $p$ for $G_{n,p}$, and $n^{-1}$ for $G_{n,m}$. This gives a (partial) solution to a problem raised by Krivelevich and Sudakov.

The formula for the asymptotic mean involves a mysterious power series.

For fixed positive integers $k,q,r$ with $q$ a prime power and large $m$, we investigate matrices with $m$ rows and a maximum number $N_q (m,k,r)$ of columns, such that each column contains at most $r$ nonzero entries from the finite field $GF(q)$ and any $k$ columns are linearly independent over $GF(q)$. For even integers $k \geq 2$ we obtain the lower bounds $N_q(m,k,r) = \Omega (m^{kr/(2(k-1))})$, and $N_q(m,k,r) = \Omega (m^{((k-1)r)/(2(k-2))})$ for odd $k \geq 3$. For $k=2^i$ we show that $N_q(m,k,r) = \Theta ( m^{kr/(2(k-1))})$ if $\gcd(k-1,r) = k-1$, while for arbitrary even $k \geq 4$ with $\gcd(k-1,r) =1$ we have $N_q(m,k,r) = \Omega (m^{kr/(2(k-1))} \cdot (\log m)^{1/(k-1)})$. Matrices which fulfil these lower bounds can be found in polynomial time. Moreover, for $\Char (GF(q)) > 2 $ we obtain $N_q(m,4,r) = \Theta (m^{\lceil 4r/3\rceil/2})$, while for $\Char (GF(q)) = 2$ we can only show that $N_q(m,4,r) = O (m^{\lceil 4r/3\rceil/2})$. Our results extend and complement earlier results from [7, 18], where the case $q=2$ was considered.

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.