The acquaintance time of (percolated) random geometric graphs

In this paper, we study the acquaintance time \AC(G) defined for a connected graph $G$. We focus on \G(n,r,p), a random subgraph of a random geometric graph in which $n$ vertices are chosen uniformly at random and independently from $[0,1]^2$, and two vertices are adjacent with probability $p$ if the Euclidean distance between them is at most $r$. We present asymptotic results for the acquaintance time of \G(n,r,p) for a wide range of $p=p(n)$ and $r=r(n)$. In particular, we show that with high probability \AC(G) = \Theta(r^{-2}) for G \in \G(n,r,1), the "ordinary" random geometric graph, provided that $\pi n r^2 - \ln n \to \infty$ (that is, above the connectivity threshold). For the percolated random geometric graph G \in \G(n,r,p), we show that with high probability \AC(G) = \Theta(r^{-2} p^{-1} \ln n), provided that p n r^2 \geq n^{1/2+\eps} and p < 1-\eps for some \eps>0

Hamilton cycles in graphs and hypergraphs: an extremal perspective

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio

Rainbow perfect matchings and Hamilton cycles in the random geometric graph

Given a graph on n vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length n visiting each vertex once and with pairwise different colours on the edges. Similarly (for even n) a rainbow perfect matching is a collection of independent edges with pairwise different colours. In this note we show that if we randomly colour the edges of a random geometric graph with sufficiently many colours, then a.a.s. the graph contains a rainbow perfect matching (rainbow Hamilton cycle) if and only if the minimum degree is at least 1 (respectively, at least 2). More precisely, consider n points (i.e. vertices) chosen independently and uniformly at random from the unit d‐dimensional cube for any fixed . Form a sequence of graphs on these n vertices by adding edges one by one between each possible pair of vertices. Edges are added in increasing order of lengths (measured with respect to the norm, for any fixed ). Each time a new edge is added, it receives a random colour chosen uniformly at random and with repetition from a set of colours, where a sufficiently large fixed constant. Then, a.a.s. the first graph in the sequence with minimum degree at least 1 must contain a rainbow perfect matching (for even n), and the first graph with minimum degree at least 2 must contain a rainbow Hamilton cycle