The spread of a connected graph G was introduced by Alon, Boppana and Spencer [1], and measures how tightly connected the graph is. It is defined as the maximum over all Lipschitz functions f on V(G) of the variance of f(X) when X is uniformly distributed on V(G). We investigate the spread for certain models of sparse random graph, in particular for random regular graphs G(n,d), for Erdős–Rényi random graphs Gn,p in the supercritical range p>1/n, and for a ‘small world’ model. For supercritical Gn,p, we show that if p=c/n with c>1 fixed, then with high probability the spread of the giant component is bounded, and we prove corresponding statements for other models of random graphs, including a model with random edge lengths. We also give lower bounds on the spread for the barely supercritical case when p=(1+o(1))/n. Further, we show that for d large, with high probability the spread of G(n,d) becomes arbitrarily close to that of the complete graph $\mathsf{K}_n$.

Bounded-size rules (BSRs) are dynamic random graph processes which incorporate limited choice along with randomness in the evolution of the system. Typically one starts with the empty graph and at each stage two edges are chosen uniformly at random. One of the two edges is then placed into the system according to a decision rule based on the sizes of the components containing the four vertices. For bounded-size rules, all components of size greater than some fixed K ≥ 1 are accorded the same treatment. Writing BSR(t) for the state of the system with ⌊nt/2⌋ edges, Spencer and Wormald [26] proved that for such rules, there exists a (rule-dependent) critical time tc such that when t < tc the size of the largest component is of order log n, while for t > tc, the size of the largest component is of order n. In this work we obtain upper bounds (that hold with high probability) of order n2γ log4n, on the size of the largest component, at time instants tn = tc−n−γ, where γ ∈ (0,1/4). This result for the barely subcritical regime forms a key ingredient in the study undertaken in [4], of the asymptotic dynamic behaviour of the process describing the vector of component sizes and associated complexity of the components for such random graph models in the critical scaling window. The proof uses a coupling of BSR processes with a certain family of inhomogeneous random graphs with vertices in the type space $\mathbb{R}_+\times \mathcal{D}([0,\infty):\mathbb{N}_0)$, where $\mathcal{D}([0,\infty):\mathbb{N}_0)$ is the Skorokhod D-space of functions that are right continuous and have left limits, with values in the space of non-negative integers $\mathbb{N}_0$, equipped with the usual Skorokhod topology. The coupling construction also gives an alternative characterization (from the usual explosion time of the susceptibility function) of the critical time tc for the emergence of the giant component in terms of the operator norm of integral operators on certain L2 spaces.

Let G be a triangle-free graph with n vertices and average degree t. We show that G contains at least

${\exp\biggl({1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t} \biggl(\frac{1}{2}\ln t-1\biggr)\biggr)}$

${e^{(c_1-o(1)) \sqrt{n} \ln n}}$

$c_1 = \sqrt{\ln 2}/4 = 0.208138 \ldots$

${e^{(c_2+o(1))\sqrt{n}\ln n}}$

$c_2 = 2\sqrt{\ln 2} = 1.665109 \ldots$

Let H be a (k+1)-uniform linear hypergraph with n vertices and average degree t. We also show that there exists a constant ck such that the number of independent sets in H is at least

${\exp\biggl({c_{k} \frac{n}{t^{1/k}}\ln^{1+1/k}{t}\biggr})}.$

$\Omega\biggl(\frac{n}{t^{1/k}} \ln^{1/k}t\biggr).$

We prove the existence of a function $f :\mathbb{N} \to \mathbb{N}$ such that the vertices of every planar graph with maximum degree Δ can be 3-coloured in such a way that each monochromatic component has at most f(Δ) vertices. This is best possible (the number of colours cannot be reduced and the dependence on the maximum degree cannot be avoided) and answers a question raised by Kleinberg, Motwani, Raghavan and Venkatasubramanian in 1997. Our result extends to graphs of bounded genus.

We show that the expected time for a random walk on a (multi-)graph G to traverse all m edges of G, and return to its starting point, is at most 2m2; if each edge must be traversed in both directions, the bound is 3m2. Both bounds are tight and may be applied to graphs with arbitrary edge lengths. This has interesting implications for Brownian motion on certain metric spaces, including some fractals.

We extend the conductance and canonical paths methods to the setting of general finite Markov chains, including non-reversible non-lazy walks. The new path method is used to show that a known bound for the mixing time of a lazy walk on a Cayley graph with a symmetric generating set also applies to the non-lazy non-symmetric case, often even when there is no holding probability.

We call a graph H Ramsey-unsaturated if there is an edge in the complement of H such that the Ramsey number r(H) of H does not change upon adding it to H. This notion was introduced by Balister, Lehel and Schelp in [J. Graph Theory51 (2006), pp. 22–32], where it is shown that cycles (except for C4) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle Cn, unless n is even and adding the chord creates an odd cycle.

We prove this conjecture for large cycles by showing a stronger statement. If a graph H is obtained by adding a linear number of chords to a cycle Cn, then r(H)=r(Cn), as long as the maximum degree of H is bounded, H is either bipartite (for even n) or almost bipartite (for odd n), and n is large.

This motivates us to call cycles strongly Ramsey-unsaturated. Our proof uses the regularity method.