Keywords

graph theory, graph minors, graph coloring, Hadwiger's conjecture

Abstract

Hadwiger's Conjecture from 1943 states that every graph with no Kt minor is (t-1)-colorable; it remains wide open for t ≥ 7. For positive integers t and s, let Kt-s denote the family of graphs obtained from the complete graph Kt by removing s edges. We say that a graph has no Kt-s minor if it has no H minor for every H in Kt-s. In 1971, Jakobsen proved that every graph with no K7-2 minor is 6-colorable. In this dissertation, we first study the extremal functions for K8-4 minors, K9-6 minors, and K10-12 minors. We show that every graph on n ≥ 9 vertices with at least 4.5n-12 edges has a K8-4 minor, every graph on n ≥ 9 vertices with at least 5n-14 edges has a K9-6 minor, and every graph on n ≥ 10 vertices with at least 5.5n-17.5 edges has a K10-12 minor. We then prove that every graph with no K8-4 minor is 7-colorable, every graph with no K9-6 minor is 8-colorable, and every graph with no K10-12 minor is 9-colorable. The proofs use the extremal functions as well as generalized Kempe chains of contraction-critical graphs obtained by Rolek and Song and a method for finding minors from three different clique subgraphs, originally developed by Robertson, Seymour, and Thomas in 1993 to prove Hadwiger's Conjecture for t = 6. Our main results imply that H-Hadwiger's Conjecture is true for each graph H on 8 vertices that is a subgraph of every graph in K8-4, each graph H on 9 vertices that is a subgraph of every graph in K9-6, and each graph H on 10 vertices that is a subgraph of every graph in K10-12.

Completion Date

2023

Semester

Fall

Committee Chair

Song, Zixia

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Degree Program

Mathematics

Format

application/pdf

Identifier

DP0028050

URL

https://purls.library.ucf.edu/go/DP0028050

Language

English

Release Date

December 2024

Length of Campus-only Access

1 year

Access Status

Doctoral Dissertation (Campus-only Access)

Campus Location

Orlando (Main) Campus

Download

Restricted to the UCF community until December 2024; it will then be open access.

Share

COinS
 

Accessibility Statement

This item was created or digitized prior to April 24, 2027, or is a reproduction of legacy media created before that date. It is preserved in its original, unmodified state specifically for research, reference, or historical recordkeeping. In accordance with the ADA Title II Final Rule, the University Libraries provides accessible versions of archival materials upon request. To request an accommodation for this item, please submit an accessibility request form.