On the Diameter of Random Subgraphs of Kneser Graphs

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CONFERENCE: Showcase of Undergraduate Research and Creative Endeavors (SOURCE)

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For natural numbers n and k, let G = KG(n,k) be the usual Kneser graph (whose vertices are k-sets of {1, 2, ... , n} with A ∼ B if and only if |A ∩ B| = 0). In a recent paper, it was shown that if n ≥ 3k- 1, then the diameter of G is 2; let Ƥ be the (monotone) graph property that a graph has diameter two (i.e., a graph H satisfies Ƥ (denoted H╞ Ƥ) if and only if diam(H) = 2). Now, let Gp be the usual binomial random subgraph of G. In our paper, we determine the threshold probability for G with respect to Ƥ as n approaches infinity, with ln(n) » k. That is, for n and k as described, we determine p0, so that, with high probability, Gp ╞ Ƥ if p » p0 and, with high probability, Gp does not satisfy Ƥ if p0 » p.