# A Lower Bound on the Hadwiger Number of a Random Subgraph of the Kneser Graph

## College

College of Arts and Sciences

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

## Share

COinS

Apr 21st, 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.