229 research outputs found

    Do 3n-5 Edges Force a Subdivision of K5?

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    Coordinated Science Laboratory was formerly known as Control Systems LaboratoryONR / N00014-85-K-0570 and N00014-88-K-031

    Locally Hamiltonian graphs and minimal size of maximal graphs on a surface

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    We prove that every locally Hamiltonian graph with n≥3n\ge 3 vertices and possibly with multiple edges has at least 3n−63n-6 edges with equality if and only if it triangulates the sphere. As a consequence, every edge-maximal embedding of a graph GG graph on some 2-dimensional surface Σ\Sigma (not necessarily compact) has at least 3n−63n-6 edges with equality if and only if GG also triangulates the sphere. If, in addition, GG is simple, then for each vertex vv, the cyclic ordering of the edges around vv on Σ\Sigma is the same as the clockwise or anti-clockwise orientation around vv on the sphere. If GG contains no complete graph on 4 vertices and has at least 4 vertices, then the face-boundaries are the same in the two embeddings.Comment: 8 page

    Density theorems for bipartite graphs and related Ramsey-type results

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    In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements

    Planar Induced Subgraphs of Sparse Graphs

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    We show that every graph has an induced pseudoforest of at least n−m/4.5n-m/4.5 vertices, an induced partial 2-tree of at least n−m/5n-m/5 vertices, and an induced planar subgraph of at least n−m/5.2174n-m/5.2174 vertices. These results are constructive, implying linear-time algorithms to find the respective induced subgraphs. We also show that the size of the largest KhK_h-minor-free graph in a given graph can sometimes be at most n−m/6+o(m)n-m/6+o(m).Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph Algorithms and Application
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