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

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Abstract

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

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