Bollobás and Riordan introduce a Tutte polynomial for coloured graphs and matroids in [3]. We observe that this polynomial has an expansion as a sum indexed by the subsets of the ground-set of a coloured matroid, generalizing the subset expansion of the Tutte polynomial. We also discuss similar expansions of other contraction–deletion invariants of graphs and matroids.

It is shown that the hard-core model on ${{\mathbb Z}}^d$ exhibits a phase transition at activities above some function $\lambda(d)$ which tends to zero as $d\rightarrow \infty$. More precisely, consider the usual nearest neighbour graph on ${{\mathbb Z}}^d$, and write ${\cal E}$ and ${\cal O}$ for the sets of even and odd vertices (defined in the obvious way). Set $${\cal G}L_M={\cal G}L_M^d =\{z\in{{\mathbb Z}}^d:\|z\|_{\infty}\leq M\},\quad \partial^{\star} {\cal G}L_M =\{z\in{{\mathbb Z}}^d:\|z\|_{\infty}= M\},$$ and write ${\cal I}({\cal G}L_M)$ for the collection of independent sets (sets of vertices spanning no edges) in ${\cal G}L_M$. For $\lambda>0$ let ${\bf I}$ be chosen from ${\cal I}({\cal G}L_M)$ with $\Pr({\bf I}=I) \propto \lambda^{|I|}$.

TheoremThere is a constant$C$such that if$\lambda > Cd^{-1/4}\log^{3/4}d$, then$$\lim_{M\rightarrow\infty}\Pr(\underline{0}\in{\bf I}|{\bf I}\supseteq \partial^{\star} {\cal G}L_M\cap {\cal E})~> \lim_{M\rightarrow\infty}\Pr(\underline{0}\in{\bf I}| {\bf I}\supseteq \partial^{\star} {\cal G}L_M\cap {\cal O}).$$ Thus, roughly speaking, the influence of the boundary on behaviour at the origin persists as the boundary recedes.

We consider two interrelated tasks in a synchronous $n$-node ring: distributed constant colouring and local communication. We investigate the impact of the amount of knowledge available to nodes on the time of completing these tasks. Every node knows the labels of nodes up to a distance $r$ from it, called the knowledge radius. In distributed constant colouring every node has to assign itself one out of a constant number of colours, so that adjacent nodes get different colours. In local communication every node has to communicate a message to both of its neighbours. We study these problems in two popular communication models: the one-way model, in which each node can only either transmit to one neighbour or receive from one neighbour, in any round, and the radio model, in which simultaneous receiving from two neighbours results in interference noise. Hence the main problem in fast execution of the above tasks is breaking symmetry with restricted knowledge of the ring.

We show that distributed constant colouring and local communication are tightly related and one can be used to accomplish the other. Also, in most situations the optimal time is the same for both of them, and it strongly depends on knowledge radius. For knowledge radius $r=0$, i.e., when each node knows only its own label, our bounds on time for both tasks are tight in both models: the optimal time in the one-way model is $\Theta(n)$, while in the radio model it is $\Theta(\log n)$. For knowledge radius $r=1$ both tasks can be accomplished in time $O(\log \log n)$ in the one-way model, if the ring is oriented. For $2 \leq r \leq c \log ^* n$, where $c < 1/2$, the upper bounds on time are $O(\log^{(2r)} n)$ in the one-way model and $O(\log ^{(2\lfloor r/2 \rfloor)} n)$ in the radio model; the lower bound is $\Omega (\log^* n)$, in both models. For $r \geq (\log^*n)/2$ both tasks can be completed in constant time, in the one-way model, and distributed constant colouring also in the radio model. Finally, if $r \geq \log^*n$ then constant time is also enough for local communication in the radio model.

We consider random planar graphs on $n$ labelled nodes, and show in particular that if the graph is picked uniformly at random then the expected number of edges is at least $\frac{13}{7}n +o(n)$. To prove this result we give a lower bound on the size of the set of edges that can be added to a planar graph on $n$ nodes and $m$ edges while keeping it planar, and in particular we see that if $m$ is at most $\frac{13}{7}n - c$ (for a suitable constant~$c$) then at least this number of edges can be added.