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The 25th British Combinatorial Conference is to take place at the University of Warwick in July 2015. The British Combinatorial Committee has invited nine distinguished combinatorialists to give survey lectures in areas of their expertise. This volume contains the survey articles on which these lectures are to be based.
We would like to thank the authors for preparing the excellent and interesting surveys, and the anonymous referees for careful reading, in some cases under tight time constraints. Also, we are grateful to the team at the Cambridge University Press, in particular Sam Harrison, for their help, advice, and professionalism. Finally, our task was much simpler thanks to the experience of the editors of earlier Surveys and the guidance of the British Combinatorial Committee.
This volume contains nine survey articles based on the invited lectures given at the 25th British Combinatorial Conference, held at the University of Warwick in July 2015. This biennial conference is a well-established international event, with speakers from around the world. The volume provides an up-to-date overview of current research in several areas of combinatorics, including graph theory, Ramsey theory, combinatorial geometry and curves over finite fields. Each article is clearly written and assumes little prior knowledge on the part of the reader. The authors are some of the world's foremost researchers in their fields, and here they summarise existing results and give a unique preview of cutting-edge developments. The book provides a valuable survey of the present state of knowledge in combinatorics, and will be useful to researchers and advanced graduate students, primarily in mathematics but also in computer science and statistics.
We consider the problem of testing expansion in bounded-degree graphs. We focus on the notion of vertex expansion: an α-expander is a graph G = (V, E) in which every subset U ⊆ V of at most |V|/2 vertices has a neighbourhood of size at least α ⋅ |U|. Our main result is that one can distinguish good expanders from graphs that are far from being weak expanders in time . We prove that the property-testing algorithm proposed by Goldreich and Ron with appropriately set parameters accepts every α-expander with probability at least and rejects every graph that is ϵ-far from any α*-expander with probability at least , where and d is the maximum degree of the graphs. The algorithm assumes the bounded-degree graphs model with adjacency list graph representation and its running time is .
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