A Lower Bound on the Hadwiger Number of a Random Subgraph of the Kneser Graph
College
College of Arts and Sciences
Department
Mathematics
Abstract
For n, k in the natural numbers, 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 = ∅). The Hadwiger number of a graph H, denoted h(H), is max{t : Kt is a minor of H}. In “Lower Bound of the Hadwiger Number of Graphs by their Average Degree,” Kostochka gives a lower bound on t with respect to the average degree of a graph H. Now, let Gp be the usual binomial random subgraph of G. In this paper, we give a general lower bound on h(Gp) as n → ∞ with k << √n; indeed, if k and p satisfy certain conditions, our lower bound is larger than previous lower bounds.
Honors Thesis Committee
Arran Hamm, Ph.D.; Kristen Abernathy, Ph.D.; and Thomas Polaski, Ph.D.
Previously Presented/Performed?
Regional Mathematics and Statistics Conference, University of North Carolina, Greensboro, November 2016; Joint Mathematics Meeting (JMM), Atlanta, Georgia, January 2017
Grant Support?
Supported by a grant from the National Institutes of Health IDeA Networks for Biomedical Research Excellence (NIH-INBRE)
Start Date
21-4-2017 3:00 PM
A Lower Bound on the Hadwiger Number of a Random Subgraph of the Kneser Graph
DiGiorgio Campus Center, Room 220
For n, k in the natural numbers, 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 = ∅). The Hadwiger number of a graph H, denoted h(H), is max{t : Kt is a minor of H}. In “Lower Bound of the Hadwiger Number of Graphs by their Average Degree,” Kostochka gives a lower bound on t with respect to the average degree of a graph H. Now, let Gp be the usual binomial random subgraph of G. In this paper, we give a general lower bound on h(Gp) as n → ∞ with k << √n; indeed, if k and p satisfy certain conditions, our lower bound is larger than previous lower bounds.
