For a fixed permutation τ, let $\mathcal{S}_N(\tau)$ be the set of permutations on N elements that avoid the pattern τ. Madras and Liu (2010) conjectured that $\lim_{N\rightarrow\infty}\frac{|\mathcal{S}_{N+1}(\tau)|}{ |\mathcal{S}_N(\tau)|}$ exists; if it does, it must equal the Stanley–Wilf limit. We prove the conjecture for every permutation τ of length 5 or less, as well as for some longer cases (including 704 of the 720 permutations of length 6). We also consider permutations drawn at random from $\mathcal{S}_N(\tau)$, and we investigate properties of their graphs (viewing permutations as functions on {1,. . .,N}) scaled down to the unit square [0,1]2. We prove exact large deviation results for these graphs when τ has length 3; it follows, for example, that it is exponentially unlikely for a random 312-avoiding permutation to have points above the diagonal strip |y−x| < ε, but not unlikely to have points below the strip. For general τ, we show that some neighbourhood of the upper left corner of [0,1]2 is exponentially unlikely to contain a point of the graph if and only if τ starts with its largest element. For patterns such as τ=4231 we establish that this neighbourhood can be extended along the sides of [0,1]2 to come arbitrarily close to the corner points (0,0) and (1,1), as simulations had suggested.

We show that asymptotically almost surely a tree with m edges decomposes the complete bipartite graph K2m,2m, a result connected to a conjecture of Graham and Häggkvist. The result also implies that asymptotically almost surely a tree with m edges decomposes the complete graph with O(m2) edges. An ingredient of the proof consists in showing that the bipartition classes of the base tree of a random tree have roughly equal size.

We study the conflict-free chromatic number χCF of graphs from extremal and probabilistic points of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erdős–Rényi random graph G(n,p) and give the asymptotics for p = ω(1/n). We also show that for p ≥ 1/2 the conflict-free chromatic number differs from the domination number by at most 3.

In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n-bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef, Birk, Jayram and Kol, IEEE Trans. Inform. Theory, 2011).

We show a polynomial-time algorithm that, given an n-vertex graph G with minrank k, finds a linear index code for G of length Õ(nf(k)), where f(k) depends only on k. For example, for k = 3 we obtain f(3) ≈ 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan for graph colouring (J. Assoc. Comput. Mach., 1998) and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank.

At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovász ϑ-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.

The d-dimensional Hamming torus is the graph whose vertices are all of the integer points inside an a1n × a2n × ⋅⋅⋅ × adn box in $\mathbb{R}^d$ (for constants a1, . . ., ad > 0), and whose edges connect all vertices within Hamming distance one. We study the size of the largest connected component of the subgraph generated by independently removing each vertex of the Hamming torus with probability 1 − p. We show that if p = λ/n, then there exists λc > 0, which is the positive root of a degree d polynomial whose coefficients depend on a1, . . ., ad, such that for λ < λc the largest component has O(log n) vertices (w.h.p. as n → ∞), and for λ > λc the largest component has $(1-q) \lambda \bigl(\prod_i a_i \bigr) n^{d-1} + o (n^{d-1})$ vertices and the second largest component has O(log n) vertices w.h.p. An implicit formula for q < 1 is also given. The value of λc that we find is distinct from the critical value for the emergence of a giant component in bond percolation on the Hamming torus.

We study a discrete time self-interacting random process on graphs, which we call greedy random walk. The walker is located initially at some vertex. As time evolves, each vertex maintains the set of adjacent edges touching it that have not yet been crossed by the walker. At each step, the walker, being at some vertex, picks an adjacent edge among the edges that have not traversed thus far according to some (deterministic or randomized) rule. If all the adjacent edges have already been traversed, then an adjacent edge is chosen uniformly at random. After picking an edge the walker jumps along it to the neighbouring vertex. We show that the expected edge cover time of the greedy random walk is linear in the number of edges for certain natural families of graphs. Examples of such graphs include the complete graph, even degree expanders of logarithmic girth, and the hypercube graph. We also show that GRW is transient in $\mathbb{Z}^d$ for all d ≥ 3.

We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t. We obtain two upper bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2-ε)Δt for graphs of maximum degree at most Δ, where ε is some absolute positive constant independent of t. The other is a bound of O(Δt/log Δ) (as Δ → ∞) for graphs of maximum degree at most Δ and girth at least 2t+1. The first bound is an analogue of Molloy and Reed's bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least g, for every fixed g ≥ 3, of arbitrarily large maximum degree Δ, with distance-t chromatic index at least Ω(Δt/log Δ).

We are given a graph G with n vertices, where a random subset of k vertices has been made into a clique, and the remaining edges are chosen independently with probability $\frac12$. This random graph model is denoted $G(n,\frac12,k)$. The hidden clique problem is to design an algorithm that finds the k-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov [3] uses spectral techniques to find the hidden clique with high probability when $k = c \sqrt{n}$ for a sufficiently large constant c > 0. Recently, an algorithm that solves the same problem was proposed by Feige and Ron [12]. It has the advantages of being simpler and more intuitive, and of an improved running time of O(n2). However, the analysis in [12] gives a success probability of only 2/3. In this paper we present a new algorithm for finding hidden cliques that both runs in time O(n2) (that is, linear in the size of the input) and has a failure probability that tends to 0 as n tends to ∞. We develop this algorithm in the more general setting where the clique is replaced by a dense random graph.

We consider irreducible Markov chains on a finite state space. We show that the mixing time of any such chain is equivalent to the maximum, over initial states x and moving large sets (As)s, of the hitting time of (As)s starting from x. We prove that in the case of the d-dimensional torus the maximum hitting time of moving targets is equal to the maximum hitting time of stationary targets. Nevertheless, we construct a transitive graph where these two quantities are not equal, resolving an open question of Aldous and Fill on a ‘cat and mouse’ game.

We study the (1:b) Maker–Breaker component game, played on the edge set of a d-regular graph. Maker's aim in this game is to build a large connected component, while Breaker's aim is to prevent him from doing so. For all values of Breaker's bias b, we determine whether Breaker wins (on any d-regular graph) or Maker wins (on almost every d-regular graph) and provide explicit winning strategies for both players.

To this end, we prove an extension of a theorem of Gallai, Hasse, Roy and Vitaver about graph orientations without long directed simple paths.

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.