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**11 - 12**of**12**### Monotone expanders- constructions and applications

"... The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. Constant degree ..."

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The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. Constant degree dimension expanders in finite fields, resolving a question of [BISW04]. 2. O(1)-page and O(1)-pushdown expanders, resolving a question of [GKS86], and leading to tight lower bounds on simulation time for certain Turing Machines. Bourgain [Bou09] gave recently an ingenious construction of such constant degree monotone expanders. The first application (1) above follows from a reduction in [DS08]. We give a short exposition of both construction and reduction. The new contributions of this paper are simple. First, we explain the observation leading to the second application (2) above, and some of its consequences. Second, we observe that a variant of the zig-zag graph product preserves monotonicity, and use it to give a simple alternative construction of monotone expanders, with near-constant degree. 1

### The Pagenumber of Genus g Graphs is 0(g)

, 1992

"... In 1979, Bernhart and Kainen conjectured that graphs of fixed genus g> 1 have unbounded pagenumber. In this paper. it is proven that genus g graphs can be embedded in 0(g) pages, thus disproving the conjecture. An 0 ( fi) lower bound is also derwed. The first algorithm in the literature for embeddi ..."

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In 1979, Bernhart and Kainen conjectured that graphs of fixed genus g> 1 have unbounded pagenumber. In this paper. it is proven that genus g graphs can be embedded in 0(g) pages, thus disproving the conjecture. An 0 ( fi) lower bound is also derwed. The first algorithm in the literature for embedding an arbitra ~ graph in a book with a non-trlwal upper bound on the number of pages M presented. First, the algorithm computes the genus g of a graph using the algorithm of Filotti, Miller, Reif ( 1979), which is polynomial-time for fixed genus. Second, it applies an optimal-time algorithm for obtaining an 0 ( g)-page book embedding. Separate book embedding algorithms are given for the cases of graphs embedded m orlentable and nonorientable surfaces. An important aspect of the construction is a new decomposition theorem, of independent interest, for a graph embedded on a surface. Book embedding has application in several areas, two of which are directly related to the results obtained: fault-tolerant VLSI and complexity theory,