229 research outputs found
Do 3n-5 Edges Force a Subdivision of K5?
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
We prove that every locally Hamiltonian graph with vertices and
possibly with multiple edges has at least edges with equality if and
only if it triangulates the sphere. As a consequence, every edge-maximal
embedding of a graph graph on some 2-dimensional surface (not
necessarily compact) has at least edges with equality if and only if
also triangulates the sphere. If, in addition, is simple, then for each
vertex , the cyclic ordering of the edges around on is the same
as the clockwise or anti-clockwise orientation around on the sphere. If
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
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
We show that every graph has an induced pseudoforest of at least
vertices, an induced partial 2-tree of at least vertices, and an
induced planar subgraph of at least vertices. These results are
constructive, implying linear-time algorithms to find the respective induced
subgraphs. We also show that the size of the largest -minor-free graph in
a given graph can sometimes be at most .Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph
Algorithms and Application
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