Influences of monotone Boolean functions

Recently, Keller and Pilpel conjectured that the influence of a monotone Boolean function does not decrease if we apply to it an invertible linear transformation. Our aim in this short note is to prove this conjecture.Comment: 3 page

Edge-disjoint Hamilton cycles in graphs

In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree \delta must have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every \alpha > 0, every (almost) regular and sufficiently large graph on n vertices with minimum degree at least $(1/2 + \alpha)n$ can be almost decomposed into edge-disjoint Hamilton cycles.Comment: Minor Revisio

The range of thresholds for diameter 2 in random Cayley graphs

Given a group G, the model G(G,p) denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. Given a family of groups (G_k) and a $c \in \mathbb{R}_+$ we say that c is the threshold for diameter 2 for (G_k) if for any ε > 0 with high probability $\Gamma \in \mathcal{G}(G_k,p)$ has diameter greater than 2 if p \leqslant \sqrt{(c - \eps)\frac{\log{n}}{n}} and diameter at most 2 if p \geqslant \sqrt{(c + \eps)\frac{\log{n}}{n}}. In [5] we proved that if c is a threshold for diameter 2 for a family of groups (G_k) then $c \in [1/4,2]$ and provided two families of groups with thresholds 1/4 and 2 respectively. In this paper we study the question of whether every $c \in [1/4,2]$ is the threshold for diameter 2 for some family of groups. Rather surprisingly it turns out that the answer to this question is negative. We show that every $c \in [1/4,4/3]$ is a threshold but a $c \in (4/3,2]$ is a threshold if and only if it is of the form 4n/(3n-1) for some positive integer n

A semi-exact degree condition for Hamilton cycles in digraphs

The paper is concerned with directed versions of Posa's theorem and Chvatal's theorem on Hamilton cycles in graphs. We show that for each a>0, every digraph G of sufficiently large order n whose outdegree and indegree sequences d_1^+ \leq ... \leq d_n^+ and d_1^- \leq >... \leq d_n^- satisfy d_i^+, d_i^- \geq min{i + a n, n/2} is Hamiltonian. In fact, we can weaken these assumptions to (i) d_i^+ \geq min{i + a n, n/2} or d^-_{n - i - a n} \geq n-i; (ii) d_i^- \geq min{i + a n, n/2} or d^+_{n - i - a n} \geq n-i; and still deduce that G is Hamiltonian. This provides an approximate version of a conjecture of Nash-Williams from 1975 and improves a previous result of K\"uhn, Osthus and Treglown

First-Order Convergence and Roots

Nesetril and Ossona de Mendez introduced the notion of first order convergence, which unifies the notions of convergence for sparse and dense graphs. They asked whether if $(G_i)_{i\in\mathbb{N}}$ is a sequence of graphs with M being their first order limit and v is a vertex of M, then there exists a sequence $(v_i)_{i\in\mathbb{N}}$ of vertices such that the graphs G_i rooted at v_i converge to M rooted at v. We show that this holds for almost all vertices v of M and we give an example showing that the statement need not hold for all vertices