# 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* : *K _{t}* 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

*G*be the usual binomial random subgraph of

_{p}*G*. In this paper, we give a general lower bound on

*h(G*as

_{p})*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* : *K _{t}* 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

*G*be the usual binomial random subgraph of

_{p}*G*. In this paper, we give a general lower bound on

*h(G*as

_{p})*n*→ ∞ with

*k*<< √

*n*; indeed, if

*k*and

*p*satisfy certain conditions, our lower bound is larger than previous lower bounds.