Let $H$ be a fixed graph on $h$ vertices. We say that a graph $G$ is induced$H$-free if it does not contain any induced copy of $H$. Let $G$ be a graph on $n$ vertices and suppose that at least $\epsilon n^2$ edges have to be added to or removed from it in order to make it induced $H$-free. It was shown in [5] that in this case $G$ contains at least $f(\epsilon,h)n^h$ induced copies of $H$, where $1/f(\epsilon,h)$ is an extremely fast growing function in $1/\epsilon$, that is independent of $n$. As a consequence, it follows that for every $H$, testing induced $H$-freeness with one-sided error has query complexity independent of $n$. A natural question, raised by the first author in [1], is to decide for which graphs $H$ the function $1/f(\epsilon,H)$ can be bounded from above by a polynomial in $1/\epsilon$. An equivalent question is: For which graphs $H$ can one design a one-sided error property tester for testing induced $H$-freeness, whose query complexity is polynomial in $1/\epsilon$? We settle this question almost completely by showing that, quite surprisingly, for any graph other than the paths of lengths 1,2 and 3, the cycle of length 4, and their complements, no such property tester exists. We further show that a similar result also applies to the case of directed graphs, thus answering a question raised by the authors in [9]. We finally show that the same results hold even in the case of two-sided error property testers. The proofs combine combinatorial, graph-theoretic and probabilistic arguments with results from additive number theory.

The theorems of Hindman and van der Waerden belong to the classical theorems of partition Ramsey Theory. The Central Sets Theorem is a strong simultaneous extension of both theorems that applies to general commutative semigroups. We give a common extension of the Central Sets Theorem and Ramsey's theorem.

We analyse Jim Propp's $P$-machine, a simple deterministic process that simulates a random walk on ${\mathbb Z}^d$ to within a constant. The proof of the error bound relies on several estimates in the theory of simple random walks and some careful summing. We mention three intriguing conjectures concerning sign-changes and unimodality of functions in the linear span of $\{p(\cdot,{\bf x}) : {\bf x} \in {\mathbb Z}^d\}$, where $p(n,{\bf x})$ is the probability that a walk beginning from the origin arrives at ${\bf x}$ at time $n$.

Every node of an undirected connected graph is coloured white or black. Adjacent nodes can be compared and the outcome of each comparison is either 0 (same colour) or 1 (different colours). The aim is to discover a node of the majority colour, or to conclude that there is the same number of black and white nodes. We consider randomized algorithms for this task and establish upper and lower bounds on their expected running time. Our main contribution are lower bounds showing that some simple and natural algorithms for this problem cannot be improved in general.

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.

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.

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 $S$.

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 \[ e(V_{1},V_{2}) > (1/2+\beta) e (G).\]

We derive the distribution of the number of links and the average weight for the shortest path tree (SPT) rooted at an arbitrary node to $m$ uniformly chosen nodes in the complete graph of size $N$ with i.i.d. exponential link weights. We rely on the fact that the full shortest path tree to all destinations (ie, $m=N-1$) is a uniform recursive tree to derive a recursion for the generating function of the number of links of the SPT, and solve this recursion exactly.

The explicit form of the generating function allows us to compute the expectation and variance of the size of the subtree for all $m$. We also obtain exact expressions for the average weight of the subtree.

Let $P_1, {\ldots}\,,P_k$ be $k$ vertex-disjoint paths in a graph $G$ where the ends of $P_i$ are $x_i$, and $y_i$. Let $H$ be the subgraph induced by the vertex sets of the paths. We find edge bounds $E_1(n)$, $E_2(n)$ such that:

1. if $e(H) \geq E_1(|V(H)|)$, then there exist disjoint paths $P_1', {\ldots}\,,P_k'$ where the ends of $P_i'$ are $x_i$ and $ y_i$ such that $|\bigcup_i V(P_i)| > |\bigcup_i V(P_i')|$;

2. if $e(H) \geq E_2(|V(H)|)$, then there exist disjoint paths $P_1', {\ldots}\,, P_k'$ where the ends of $P_i'$ are $x_i'$ and $y_i'$ such that $|\bigcup_i V(P_i)| > |\bigcup_i V(P_i')|$ and $\{ x_1, {\ldots}\,, x_k \} = \{ x_1' , {\ldots}\, , x_k' \}$ and $\{ y_1, {\ldots}\,, y_k \} = \{ y_1', {\ldots}\,, y_k'\}$.

The bounds are the best possible, in that there exist arbitrarily large graphs $H'$ with $e(H') = E_i (H') - 1$ without the properties stipulated in 1 and 2.

There are several known results asserting that undirected graphs can be partitioned in a way that satisfies various constraints imposed on the degrees. The corresponding results for directed graphs, where degrees are replaced by outdegrees, often fail, and when they do hold, they are usually much harder, and lead to fascinating open problems. In this note we list three problems of this type, and mention the undirected analogues. All graphs and digraphs considered here are simple, that is, they have no loops and no multiple edges.