We study relational structures (especially graphs and posets) which satisfy the analogue of homogeneity but for homomorphisms rather than isomorphisms. The picture is rather different. Our main results are partial characterizations of countable graphs and posets with this property; an analogue of Fraïssé's theorem; and representations of monoids as endomorphism monoids of such structures.

It is well known that counting $\lambda$-colourings ($\lambda\geq 3$) is #P-complete for general graphs, and also for several restricted classes such as bipartite planar graphs. On the other hand, it is known to be polynomial time computable for graphs of bounded tree-width. There is often special interest in counting colourings of square grids, and such graphs can be regarded as borderline graphs of unbounded tree-width in a specific sense. We are thus motivated to consider the complexity of counting colourings of subgraphs of the square grid. We show that the problem is #P-complete when $\lambda\geq 3$. It remains #P-complete when restricted to induced subgraphs with maximum degree 3.

The notion of conductance introduced by Jerrum and Sinclair [8] has been widely used to prove rapid mixing of Markov chains. Here we introduce a bound that extends this in two directions. First, instead of measuring the conductance of the worst subset of states, we bound the mixing time by a formula that can be thought of as a weighted average of the Jerrum–Sinclair bound (where the average is taken over subsets of states with different sizes). Furthermore, instead of just the conductance, which in graph theory terms measures edge expansion, we also take into account node expansion. Our bound is related to the logarithmic Sobolev inequalities, but it appears to be more flexible and easier to compute.

In the case of random walks in convex bodies, we show that this new bound is better than the known bounds for the worst case. This saves a factor of $O(n)$ in the mixing time bound, which is incurred in all proofs as a ‘penalty’ for a ‘bad start’. We show that in a convex body in $\mathbb{R}^n$, with diameter $D$, random walk with steps in a ball with radius $\delta$ mixes in $O^*(nD^2/\delta^2)$ time (if idle steps at the boundary are not counted). This gives an $O^*(n^3)$ sampling algorithm after appropriate preprocessing, improving the previous bound of $O^*(n^4)$.

The application of the general conductance bound in the geometric setting depends on an improved isoperimetric inequality for convex bodies.

We generalize and unify results on parametrized and coloured Tutte polynomials of graphs and matroids due to Zaslavsky, and Bollobás and Riordan. We give a generalized Zaslavsky–Bollobás–Riordan theorem that characterizes parametrized contraction–deletion functions on minor-closed classes of matroids, as well as the modifications necessary to apply the discussion to classes of graphs. In general, these parametrized Tutte polynomials do not satisfy analogues of all the familiar properties of the classical Tutte polynomial. We give conditions under which they do satisfy corank-nullity formulas, and also conditions under which they reflect the structure of series-parallel connections.

Let $p_c({\mathbb Q}_n)$ and $p_c({\mathbb Z}^n)$ denote the critical values for nearest-neighbour bond percolation on the $n$-cube ${\mathbb Q}_n = \{0,1\}^n$ and on ${\mathbb Z}^n$, respectively. Let $\Omega = n$ for ${\mathbb G} = {\mathbb Q}_n$ and $\Omega = 2n$ for ${\mathbb G} = {\mathbb Z}^n$ denote the degree of ${\mathbb G}$. We use the lace expansion to prove that for both ${\mathbb G} = {\mathbb Q}_n$ and ${\mathbb G} = {\mathbb Z}^n$, \[p_c({\mathbb G}) = \Omega^{-1} + \Omega^{-2} + \frac{7}{2} \Omega^{-3} + O(\Omega^{-4}).\] This extends by two terms the result $p_c({\mathbb Q}_n) = \Omega^{-1} + O(\Omega^{-2})$ of Borgs, Chayes, van der Hofstad, Slade and Spencer, and provides a simplified proof of a previous result of Hara and Slade for ${\mathbb Z}^n$.

Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability $p$, independently of each other, and a deterministic spreading rule with a fixed parameter $k$: if a vacant site has at least $k$ occupied neighbours at a certain time step, then it becomes occupied in the next step. This process is well studied on ${\mathbb Z}^d$; here we investigate it on regular and general infinite trees and on non-amenable Cayley graphs. The critical probability is the infimum of those values of $p$ for which the process achieves complete occupation with positive probability. On trees we find the following discontinuity: if the branching number of a tree is strictly smaller than $k$, then the critical probability is 1, while it is $1-1/k$ on the $k$-ary tree. A related result is that in any rooted tree $T$ there is a way of erasing $k$ children of the root, together with all their descendants, and repeating this for all remaining children, and so on, such that the remaining tree $T'$ has branching number $\mbox{\rm br}(T')\leq \max\{\mbox{\rm br}(T)-k,\,0\}$. We also prove that on any $2k$-regular non-amenable graph, the critical probability for the $k$-rule is strictly positive.

We show that evaluating the Tutte polynomial for the class of bicircular matroids is #P-hard at every point $(x,y)$ except those in the hyperbola $(x-1)(y-1)=1$ and possibly those on the lines $x=0$ and $x=-1$. Since bicircular matroids form a rather restricted subclass of transversal matroids, our results can be seen as a partial strengthening of a result by Colbourn, Provan and Vertigan, namely that the evaluation of the Tutte polynomial for the class of transversal matroids is #P-hard for all points except those in the hyperbola $(x-1)(y-1)=1$.

Let p and q be distinct primes. We characterize transitive groups G that admit a complete block system of q blocks of size p such that the subgroup of G which fixes each block set-wise has a Sylow p-subgroup of order p. Using this result, we prove that the full automorphism group of a metacirculant graph Γ of order pq such that Aut(Γ) is imprimitive, is contained in one of several families of transitive groups. As the automorphism groups of vertex-transitive graphs of order pq that are primitive have been determined by several authors, this result implies that automorphism groups of vertex-transitive graphs of order pq are known. We also determine all nonnormal Cayley graphs of order pq, and all 1/2-transitive graphs of order pq.

By the complexity of a graph we mean the minimum number of union and intersection operations needed to obtain the whole set of its edges starting from stars. This measure of graphs is related to the circuit complexity of boolean functions.

We prove some lower bounds on the complexity of explicitly given graphs. This yields some new lower bounds for boolean functions, as well as new proofs of some known lower bounds in the graph-theoretic framework. We also formulate several combinatorial problems whose solution would have intriguing consequences in computational complexity.

In this paper, we study percolation on finite Cayley graphs. A conjecture of Benjamini says that the critical percolation $p_c$ of any vertex-transitive graph satisfying a certain diameter condition can be bounded away from one. We prove Benjamini's conjecture for some special classes of Cayley graphs. We also establish a reduction theorem, which allows us to build Cayley graphs for large groups without increasing $p_c$.

It is a classical result of Jaeger, Vertigan and Welsh that evaluating the Tutte polynomial of a graph is #P-hard in all but a few special points. On the other hand, several papers in the past few years have shown that the Tutte polynomial of a graph can be efficiently computed for graphs of bounded tree-width. In this paper we present a recursive formula computing the Tutte polynomial of a matroid $M$ represented over a finite field (which includes all graphic matroids), using a so called parse tree of a branch-decomposition of $M$. This formula provides an algorithm computing the Tutte polynomial for a representable matroid of bounded branch-width in polynomial time with a fixed exponent.

For any partition of $\{1, 2, \ldots{,}\, n\}$ we define its increments$X_i, 1 \leq i \leq n$ by $X_i = 1$ if $i$ is the smallest element in the partition block that contains it, $X_i = 0$ otherwise. We prove that for partially exchangeable random partitions (where the probability of a partition depends only on its block sizes in order of appearance), the law of the increments uniquely determines the law of the partition. One consequence is that the Chinese Restaurant Process CRP($\theta$) (the partition with distribution given by the Ewens sampling formula with parameter $\theta$) is the only exchangeable random partition with independent increments.

Polyhedral embeddings of cubic graphs by means of certain operations are studied. It is proved that some known families of snarks have no (orientable) polyhedral embeddings. This result supports a conjecture of Grünbaum that no snark admits an orientable polyhedral embedding. This conjecture is verified by computer for all snarks having fewer than 30 vertices. On the other hand, for every non-orientable surface $S$, there exists a non-3-edge-colourable graph which polyhedrally embeds in $S$.

Let $\cal{B}(n, \leq 4)$ denote the subsets of $[n]:=\{ 1, 2, \dots, n\}$ of at most 4 elements. Suppose that $\cal{F}$ is a set system with the property that every member of $\cal{B}$ can be written as a union of (at most) two members of $\cal{F}$. (Such an $\cal{F}$ is called a 2-base of $\cal{B}$.) Here we answer a question of Erdős proving that \[|\FF|\geq 1+n+\binom{n}{2}- \Bigl\lfloor \frac{4}{3}n\Bigr\rfloor\], and this bound is best possible for $n\geq 8$.

We consider instances of the maximum independent set problem that are constructed according to the following semirandom model. Let $G_{n,p}$ be a random graph, and let $S$ be a set of $k$ vertices, chosen uniformly at random. Then, let $G_0$ be the graph obtained by deleting all edges connecting two vertices in $S$. Finally, an adversary may add edges to $G_0$ that do not connect two vertices in $S$, thereby producing the instance $G=G_{n,p,k}^*$. We present an algorithm that on input $G=G_{n,p,k}^*$ finds an independent set of size $\geq k$ within polynomial expected time, provided that $k\geq C(n/p)^{1/2}$ for a certain constant $C>0$. Moreover, we prove that in the case $k\leq (1-\varepsilon)\ln(n)/p$ this problem is hard.

We will study the best way to reveal a hidden perfect matching in a balanced bipartite graph by eliminating edges, one by one, in the hope that the eliminated edge is not part of the mystery perfect matching. We will look for the strategy that maximizes the odds of finding the perfect matching without revealing a fixed number of the edges in that perfect matching. For a complete bipartite graph, this is equivalent to finding a mystery permutation via negative guesses with only a fixed number of incorrect negative guesses.

We present an elementary proof that if $A$ is a finite set of numbers, and the sumset $A+_GA$ is small, $|A+_GA|\leq c|A|$, along a dense graph $G$, then $A$ contains $k$-term arithmetic progressions.

The main results of this paper are regularity and counting lemmas for 3-uniform hypergraphs. A combination of these two results gives a new proof of a theorem of Frankl and Rödl, of which Szemerédi's theorem for arithmetic progressions of length 4 is a notable consequence. Frankl and Rödl also prove regularity and counting lemmas, but the proofs here, and even the statements, are significantly different. Also included in this paper is a proof of Szemerédi's regularity lemma, some basic facts about quasirandomness for graphs and hypergraphs, and detailed explanations of the motivation for the definitions used.

Markov chain decomposition is a tool for analysing the convergence rate of a complicated Markov chain by studying its behaviour on smaller, more manageable pieces of the state space. Roughly speaking, if a Markov chain converges quickly to equilibrium when restricted to subsets of the state space, and if there is sufficient ergodic flow between the pieces, then the original Markov chain also must converge rapidly to equilibrium. We present a new version of the decomposition theorem where the pieces partition the state space, rather than forming a cover where pieces overlap, as was previously required. This new formulation is more natural and better suited to many applications. We apply this disjoint decomposition method to demonstrate the efficiency of simple Markov chains designed to uniformly sample circuits of a given length on certain Cayley graphs. The proofs further indicate that a Markov chain for sampling adsorbing staircase walks, a problem arising in statistical physics, is also rapidly mixing.

We show that if a graph contains few copies of a given graph, then its edges are distributed rather unevenly.

In particular, for all $\varepsilon > 0$ and $r\geq2$, there exist $\xi =\xi (\varepsilon,r) > 0$ and $k=k (\varepsilon,r)$ such that, if $n$ is sufficiently large and $G=G(n)$ is a graph with fewer than $\xi n^{r}$$r$-cliques, then there exists a partition $V(G) =\cup_{i=0}^{k}V_{i}$ such that \[ \vert V_{i}\vert =\lfloor n/k\rfloor \quad \text{and} \quad e(W_{i}) <\varepsilon\vert V_{i}\vert ^{2}\] for every $i\in [k]$.

We deduce the following slightly stronger form of a conjecture of Erdős.

For all $c>0$ and $r\geq3$, there exist $\xi=\xi (c,r) >0$ and $\beta=\beta(c,r)>0$ such that, if $n$ is sufficiently large and $G=G(n,\lceil cn^{2} \rceil)$ is a graph with fewer than $\xi n^{r}$$r$-cliques, then there exists a partition $V(G) =V_{1}\cup V_{2}$ with $ \vert V_{1} \vert = \lfloor n/2 \rfloor $ and $\vert V_{2} \vert = \lceil n/2 \rceil $ such that \[ e(V_{1},V_{2}) > (1/2+\beta) e (G).\]