#### Event Title

On the Diameter of Random Subgraphs of Kneser Graphs

#### College

College of Arts and Sciences

#### Department

Department of Mathematics

#### Honors Thesis Committee

Arran Hamm, Ph.D.; Thomas Polaski, Ph.D.; and Jessica Hamm, Ph.D.

#### Location

West Center, Room 219

#### Start Date

21-4-2017 12:45 PM

#### Description

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 *≥ 3*k*- 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 *G _{p}* 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

*p*, so that, with high probability,

_{0}*G*╞

_{p}*Ƥ*if

*p*»

*p*and, with high probability,

_{0}*G*does not satisfy

_{p}*Ƥ*if

*p*»

_{0}*p*.

On the Diameter of Random Subgraphs of Kneser Graphs

West Center, Room 219

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 *≥ 3*k*- 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 *G _{p}* 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

*p*, so that, with high probability,

_{0}*G*╞

_{p}*Ƥ*if

*p*»

*p*and, with high probability,

_{0}*G*does not satisfy

_{p}*Ƥ*if

*p*»

_{0}*p*.