5,488 research outputs found

    The history of degenerate (bipartite) extremal graph problems

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    This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

    Breaking Symmetries in Graph Representation

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    There are many complex combinatorial problems which involve searching for an undirected graph satisfying a certain property. These problems are often highly challenging because of the large number of isomorphic representations of a possible solution. In this paper we introduce novel, effective and compact, symmetry breaking constraints for undirected graph search. While incomplete, these prove highly beneficial in pruning the search for a graph. We illustrate the application of symmetry breaking in graph representation to resolve several open instances in extremal graph theory

    Extremal higher codimension cycles on moduli spaces of curves

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    We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli space Mg,n\overline{\mathcal M}_{g,n} of stable genus gg curves with nn ordered marked points. In particular, we prove that codimension two boundary strata are extremal and exhibit extremal boundary strata of higher codimension. We also show that the locus of hyperelliptic curves with a marked Weierstrass point in M3,1\overline{\mathcal M}_{3,1} and the locus of hyperelliptic curves in M4\overline{\mathcal M}_4 are extremal cycles. In addition, we exhibit infinitely many extremal codimension two cycles in M1,n\overline{\mathcal M}_{1,n} for n5n\geq 5 and in M2,n\overline{\mathcal M}_{2,n} for n2n\geq 2.Comment: 25 page

    Hamilton cycles in hypergraphs below the Dirac threshold

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    We establish a precise characterisation of 44-uniform hypergraphs with minimum codegree close to n/2n/2 which contain a Hamilton 22-cycle. As an immediate corollary we identify the exact Dirac threshold for Hamilton 22-cycles in 44-uniform hypergraphs. Moreover, by derandomising the proof of our characterisation we provide a polynomial-time algorithm which, given a 44-uniform hypergraph HH with minimum codegree close to n/2n/2, either finds a Hamilton 22-cycle in HH or provides a certificate that no such cycle exists. This surprising result stands in contrast to the graph setting, in which below the Dirac threshold it is NP-hard to determine if a graph is Hamiltonian. We also consider tight Hamilton cycles in kk-uniform hypergraphs HH for k3k \geq 3, giving a series of reductions to show that it is NP-hard to determine whether a kk-uniform hypergraph HH with minimum degree δ(H)12V(H)O(1)\delta(H) \geq \frac{1}{2}|V(H)| - O(1) contains a tight Hamilton cycle. It is therefore unlikely that a similar characterisation can be obtained for tight Hamilton cycles.Comment: v2: minor revisions in response to reviewer comments, most pseudocode and details of the polynomial time reduction moved to the appendix which will not appear in the printed version of the paper. To appear in Journal of Combinatorial Theory, Series
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