1,568 research outputs found

    The random k-matching-free process

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    Let P\mathcal{P} be a graph property which is preserved by removal of edges, and consider the random graph process that starts with the empty nn-vertex graph and then adds edges one-by-one, each chosen uniformly at random subject to the constraint that P\mathcal{P} is not violated. These types of random processes have been the subject of extensive research over the last 20 years, having striking applications in extremal combinatorics, and leading to the discovery of important probabilistic tools. In this paper we consider the kk-matching-free process, where P\mathcal{P} is the property of not containing a matching of size kk. We are able to analyse the behaviour of this process for a wide range of values of kk; in particular we prove that if k=o(n)k=o(n) or if n2k=o(n/logn)n-2k=o(\sqrt{n}/\log n) then this process is likely to terminate in a kk-matching-free graph with the maximum possible number of edges, as characterised by Erd\H{o}s and Gallai. We also show that these bounds on kk are essentially best possible, and we make a first step towards understanding the behaviour of the process in the intermediate regime

    Isomorph-free generation of 2-connected graphs with applications

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    Many interesting graph families contain only 2-connected graphs, which have ear decompositions. We develop a technique to generate families of unlabeled 2-connected graphs using ear augmentations and apply this technique to two problems. In the first application, we search for uniquely K_r-saturated graphs and find the list of uniquely K_4-saturated graphs on at most 12 vertices, supporting current conjectures for this problem. In the second application, we verifying the Edge Reconstruction Conjecture for all 2-connected graphs on at most 12 vertices. This technique can be easily extended to more problems concerning 2-connected graphs.Comment: 15 pages, 3 figures, 4 table

    Reconfiguring Independent Sets in Claw-Free Graphs

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    We present a polynomial-time algorithm that, given two independent sets in a claw-free graph GG, decides whether one can be transformed into the other by a sequence of elementary steps. Each elementary step is to remove a vertex vv from the current independent set SS and to add a new vertex ww (not in SS) such that the result is again an independent set. We also consider the more restricted model where vv and ww have to be adjacent
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