72 research outputs found
Jigsaw percolation on random hypergraphs
The jigsaw percolation process on graphs was introduced by Brummitt,
Chatterjee, Dey, and Sivakoff as a model of collaborative solutions of puzzles
in social networks. Percolation in this process may be viewed as the joint
connectedness of two graphs on a common vertex set. Our aim is to extend a
result of Bollob\'as, Riordan, Slivken, and Smith concerning this process to
hypergraphs for a variety of possible definitions of connectedness. In
particular, we determine the asymptotic order of the critical threshold
probability for percolation when both hypergraphs are chosen binomially at
random.Comment: 17 page
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
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
Multi-coloured jigsaw percolation on random graphs
The jigsaw percolation process, introduced by Brummitt, Chatterjee, Dey and
Sivakoff, was inspired by a group of people collectively solving a puzzle. It
can also be seen as a measure of whether two graphs on a common vertex set are
"jointly connected". In this paper we consider the natural generalisation of
this process to an arbitrary number of graphs on the same vertex set. We prove
that if these graphs are random, then the jigsaw percolation process exhibits a
phase transition in terms of the product of the edge probabilities. This
generalises a result of Bollob\'as, Riordan, Slivken and Smith.Comment: 13 page
The hitting time of clique factors
In a recent paper, Kahn gave the strongest possible, affirmative, answer to
Shamir's problem, which had been open since the late 1970s: Let and
let be divisible by . Then, in the random -uniform hypergraph process
on vertices, as soon as the last isolated vertex disappears, a perfect
matching emerges. In the present work, we transfer this hitting time result to
the setting of clique factors in the random graph process: At the time that the
last vertex joins a copy of the complete graph , the random graph process
contains a -factor. Our proof draws on a novel sequence of couplings,
extending techniques of Riordan and the first author. An analogous result is
proved for clique factors in the -uniform hypergraph process ()
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