On the Diameter of Random Subgraphs of Kneser Graphs
 Citation data:

CONFERENCE: Showcase of Undergraduate Research and Creative Endeavors (SOURCE)
 Publication Year:
 2017

 Bepress 6
 Author(s):
conference paper description
For natural numbers n and k, let G = KG(n,k) be the usual Kneser graph (whose vertices are ksets 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.