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The arc-representation of a graph is a mapping from the set of vertices to the arcs of a circle such that adjacent vertices are mapped to intersecting arcs. The width of such a representation is the maximum number of arcs having a point in common. The arc-width(aw) of a graph is the minimum width of its arc-representations. We show how arc-width is related(More)
1 Abstract. The rst result concerns branching programs having width (log n) O(1). We give an (n log n= log log n) lower bound for the size of such branching programs computing almost any symmetric Boolean function and in particular the following explicit function: \the sum of the input variables is a quadratic residue mod p" where p is any given prime(More)
2 Abstract Dirac's classical theorem asserts that, if every vertex of a graph G on n vertices has degree at least n 2 then G has a Hamiltonian cycle. We give a fast parallel algorithm on a CREW ?PRAM to nd a Hamiltonian cycle in such graphs. Our algorithm uses a linear number of processors and is optimal up to a polylogarithmic factor. The algorithm works(More)
Islands are combinatorial objects that can be intuitively defined on a board consisting of a finite number of cells. Based on the neighbor relation of the cells, it is a fundamental property that two islands are either containing or disjoint. Recently, numerous extremal questions have been answered using different methods. We show elementary techniques(More)