3,432 research outputs found

    On the bipartite independence number of a balanced bipartite graph

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    AbstractThe bipartite independence number αBIP of a bipartite graph G is the maximum order of a balanced independent set of G. Let δ be the minimum degree of the graph. When G itself is balanced, we establish some relations between αBIP and the size or the connectivity of G. We also prove that the condition αBIP⩽δ(resp.αBIP⩽δ−1) implies that G is hamiltonian (resp. Hamilton-biconnected), thus improving a result of Fraisse

    The early evolution of the H-free process

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    The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as nn \to \infty, the minimum degree in G is at least cn1(vH2)/(eH1)(logn)1/(eH1)cn^{1-(v_H-2)/(e_H-1)}(\log n)^{1/(e_H-1)}. This gives new lower bounds for the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite graphs Kr,rK_{r,r} with r5r \ge 5. When H is a complete graph KsK_s with s5s \ge 5 we show that for some C>0, with high probability the independence number of G is at most Cn2/(s+1)(logn)11/(eH1)Cn^{2/(s+1)}(\log n)^{1-1/(e_H-1)}. This gives new lower bounds for Ramsey numbers R(s,t) for fixed s5s \ge 5 and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.Comment: 36 page

    Cohen-Macaulay graphs and face vectors of flag complexes

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    We introduce a construction on a flag complex that, by means of modifying the associated graph, generates a new flag complex whose hh-factor is the face vector of the original complex. This construction yields a vertex-decomposable, hence Cohen-Macaulay, complex. From this we get a (non-numerical) characterisation of the face vectors of flag complexes and deduce also that the face vector of a flag complex is the hh-vector of some vertex-decomposable flag complex. We conjecture that the converse of the latter is true and prove this, by means of an explicit construction, for hh-vectors of Cohen-Macaulay flag complexes arising from bipartite graphs. We also give several new characterisations of bipartite graphs with Cohen-Macaulay or Buchsbaum independence complexes.Comment: 14 pages, 3 figures; major updat

    Minimum degree conditions for monochromatic cycle partitioning

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    A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any rr-edge-coloured complete graph has a partition into O(r2logr)O(r^2 \log r) monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant cc such that any rr-edge-coloured graph on nn vertices with minimum degree at least n/2+crlognn/2 + c \cdot r \log n has a partition into O(r2)O(r^2) monochromatic cycles. We also provide constructions showing that the minimum degree condition and the number of cycles are essentially tight.Comment: 22 pages (26 including appendix

    Some recent results in the analysis of greedy algorithms for assignment problems

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    We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems
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