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We prove that the crossing number of a graph decays in a " continuous fashion " in the following sense. For any ε > 0 there is a δ > 0 such that for a sufficiently large n, every graph G with n vertices and m ≥ n 1+ε edges, has a subgraph G ′ of at most (1 − δ)m edges and crossing number at least (1 − ε)cr(G). This generalizes the result of J. Fox and Cs.(More)
Let X be a set of points in general position in the plane. General position means that no three points lie on a line and no two points have the same x-coordinate. Y ⊆ X is a cup, resp. cap, if the points of Y lie on the graph of a convex, resp. concave function. Denote the points of Y by p 1 , p 2 ,. .. , p m according to the increasing x-coordinate. The(More)
For an integer <i>h</i> &#8805; 1, an <i>elementary h-route flow</i> is a flow along <i>h</i> edge disjoint paths between a source and a sink, each path carrying a unit of flow, and a single commodity <i>h-route flow</i> is a non-negative linear combination of elementary <i>h</i>-flows. An instance of a <i>single source multicommodity flow problem</i> for a(More)
For an integer h ≥ 1, an elementary h-route flow is a flow along h edge disjoint paths between a source and a sink, each path carrying a unit of flow, and a single commodity h-route flow is a non-negative linear combination of elementary h-route flows. An instance of a single source multicommodity flow problem for a In the single source multicommodity(More)
For an integer h ≥ 1, an elementary h-route flow is a flow along h edge disjoint paths between a source and a sink, each path carrying a unit of flow, and a single commodity h-route flow is a non-negative linear combination of elementary h-route flows. An instance of a single source multicommodity flow problem for a graph G = (V, E) consists of a source(More)
For a vector field a = ∂J, where J is a convex, coercive and differentiable function on IR N without any growth conditions, we define generalized solution for the nonlinear elliptic equation −div a(∇u) = f in Ω, u = 0 on ∂Ω where Ω is a smooth bounded domain in IR N and f ∈ L 1 (Ω). We prove existence of a generalized solution and show that this solution is(More)
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