6,061 research outputs found
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
The effect of network structure on phase transitions in queuing networks
Recently, De Martino et al have presented a general framework for the study
of transportation phenomena on complex networks. One of their most significant
achievements was a deeper understanding of the phase transition from the
uncongested to the congested phase at a critical traffic load. In this paper,
we also study phase transition in transportation networks using a discrete time
random walk model. Our aim is to establish a direct connection between the
structure of the graph and the value of the critical traffic load. Applying
spectral graph theory, we show that the original results of De Martino et al
showing that the critical loading depends only on the degree sequence of the
graph -- suggesting that different graphs with the same degree sequence have
the same critical loading if all other circumstances are fixed -- is valid only
if the graph is dense enough. For sparse graphs, higher order corrections,
related to the local structure of the network, appear.Comment: 12 pages, 7 figure
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