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 $n$.

This paper is devoted to an online variant of the minimum spanning tree problem in randomly weighted graphs. We assume that the input graph is complete and the edge weights are uniformly distributed over [0,1]. An algorithm receives the edges one by one and has to decide immediately whether to include the current edge into the spanning tree or to reject it. The corresponding edge sequence is determined by some adversary. We propose an algorithm which achieves $\mathbb{E}[ALG]/\mathbb{E}[OPT]=O(1)$ and $\mathbb{E}[ALG/OPT]=O(1)$ against a fair adaptive adversary, i.e., an adversary which determines the edge order online and is fair in a sense that he does not know more about the edge weights than the algorithm. Furthermore, we prove that no online algorithm performs better than $\mathbb{E}[ALG]/\mathbb{E}[OPT]=\Omega(\log n)$ if the adversary knows the edge weights in advance. This lower bound is tight, since there is an algorithm which yields $\mathbb{E}[ALG]/\mathbb{E}[OPT]=O(\log n)$ against the strongest-imaginable adversary.

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 study negative dependence properties of a sampling process due to Srinivasan to produce distributions on level sets with given marginals. We give a simple proof that the distribution satisfies negative association. We also show that under a linear match schedule it satisfies the stronger condition of conditional negative association via a non-trivial application of the Feder–Mihail theorem. This method involves the notion of a variable of positive influence. We give some results and related counter-examples which might shed some light on its role in a theory of negative dependence.

A simple first moment argument shows that in a randomly chosen $k$-SAT formula with $m$ clauses over $n$ boolean variables, the fraction of satisfiable clauses is $1-2^{-k}+o(1)$ as $m/n\rightarrow\infty$ almost surely. In this paper, we deal with the corresponding algorithmic strong refutation problem: given a random $k$-SAT formula, can we find a certificate that the fraction of satisfiable clauses is $1-2^{-k}+o(1)$ in polynomial time? We present heuristics based on spectral techniques that in the case $k=3$ and $m\geq\ln(n)^6n^{3/2}$, and in the case $k=4$ and $m\geq Cn^2$, find such certificates almost surely. In addition, we present heuristics for bounding the independence number (resp. the chromatic number) of random $k$-uniform hypergraphs from above (resp. from below) for $k=3,4$.

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 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.

A cylinder graph is the graph Cartesian product of a path and a cycle. In this paper we investigate the length of a minimal spanning tree of a cylinder graph whose edges are assigned random lengths according to independent and uniformly distributed random variables. Our work was inspired by a formula of J. Michael Steele which shows that the expected length of a minimal spanning tree of a connected graph can be calculated through the Tutte polynomial of the graph. First, using transfer matrices, we show how to calculate the Tutte polynomials of cylinder graphs. Second, using Steele's formula, we tabulate the expected lengths of the minimal spanning trees for some cylinder graphs. Third, for a fixed cycle length, we show that the ratio of the expected length of a minimal spanning tree of a cylinder graph to the length of the cylinder graph converges to a constant; this constant is described in terms of the Perron–Frobenius eigenvalue of the accompanying transfer matrix. Finally, we show that the length of a minimal spanning tree of a cylinder graph satisfies a strong law of large numbers.