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.

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.

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 \[ \prod_{1\leq j\leq k } \sigma_j \big(v^{-1}x_ {i_j }\big) =1.\]

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].

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.

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.

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.

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].