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The Degree-Diameter Problem for Claw-Free Graphs and Hypergraphs



We study the degree-diameter problem for claw-free graphs and 2-regular hypergraphs. Let inline image be the largest order of a claw-free graph of maximum degree Δ and diameter D. We show that inline image, where inline image, for any D and any even inline image. So for claw-free graphs, the well-known Moore bound can be strengthened considerably. We further show that inline image for inline image with inline image (mod 4). We also give an upper bound on the order of inline image-free graphs of given maximum degree and diameter for inline image. We prove similar results for the hypergraph version of the degree-diameter problem. The hypergraph Moore bound states that the order of a hypergraph of maximum degree Δ, rank k, and diameter D is at most inline image. For 2-regular hypergraph of rank inline image and any diameter D, we improve this bound to inline image, where inline image. Our construction of claw-free graphs of diameter 2 yields a similar result for hypergraphs of diameter 2, degree 2, and any even rank inline image.

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