Hamiltonicity and Structure of Classes of Minor-Free Graphs
The main results of this dissertation are Hamiltonicity and structural results for graphs on surfaces and graphs with certain forbidden minors. The first result is related to a conjecture due to Grunbaum and Nash-Williams which states that all 4-connected graphs on the torus are Hamiltonian. One app...
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ndltd-VANDERBILT-oai-VANDERBILTETD-etd-03212014-1521162014-04-09T05:01:35Z Hamiltonicity and Structure of Classes of Minor-Free Graphs Marshall, Emily Abernethy Mathematics The main results of this dissertation are Hamiltonicity and structural results for graphs on surfaces and graphs with certain forbidden minors. The first result is related to a conjecture due to Grunbaum and Nash-Williams which states that all 4-connected graphs on the torus are Hamiltonian. One approach to prove this conjecture is to extend the proof techniques of a result due to Thomas and Yu which says that every edge of a 4-connected projective-planar graph is on a Hamilton cycle. However the analogous result is not true for graphs on the torus. Thomassen provided examples of 4-connected toroidal graphs such that some edges of each graph are not contained in any Hamilton cycle. Our result shows that these examples are critical in a certain sense. The second and third results concern minor-free graphs. Tutte proved that every 4-connected planar graph is Hamiltonian. Not all 3-connected planar graphs are Hamiltonian, however: the Herschel graph is one example. Our second result proves that all 3-connected, planar, K_{2,5}-minor-free graphs are Hamiltonian. We give examples to show that the K_{2,5}-minor-free condition cannot be weakened to K_{2,6}-minor-free. The final result is a complete characterization of all K_{2,4}-minor-free graphs. To prove both of these results we first provide a characterization of rooted-K_{2,2}-minor-free graphs. We also prove several useful results concerning Hamilton paths in rooted K_{2,2}-minor-free graphs. Xiaoya Zha Denis Osin Michael Mihalik Mark Ellingham Jeremy Spinrad VANDERBILT 2014-04-08 text application/pdf http://etd.library.vanderbilt.edu/available/etd-03212014-152116/ http://etd.library.vanderbilt.edu/available/etd-03212014-152116/ en unrestricted I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to Vanderbilt University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
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Mathematics Marshall, Emily Abernethy Hamiltonicity and Structure of Classes of Minor-Free Graphs |
description |
The main results of this dissertation are Hamiltonicity and structural results for graphs on surfaces and graphs with certain forbidden minors.
The first result is related to a conjecture due to Grunbaum and Nash-Williams which states that all 4-connected graphs on the torus are Hamiltonian. One approach to prove this conjecture is to extend the proof techniques of a result due to Thomas and Yu which says that every edge of a 4-connected projective-planar graph is on a Hamilton cycle. However the analogous result is not true for graphs on the torus. Thomassen
provided examples of 4-connected toroidal graphs such that some edges of each graph are not contained in any Hamilton cycle. Our result shows that these examples are critical in a certain sense.
The second and third results concern minor-free graphs. Tutte proved that every 4-connected planar
graph is Hamiltonian. Not all 3-connected planar graphs are Hamiltonian, however: the Herschel graph is one example. Our second result proves that all 3-connected, planar, K_{2,5}-minor-free graphs are Hamiltonian. We give examples to show that the K_{2,5}-minor-free condition cannot be weakened to K_{2,6}-minor-free. The final result is a complete characterization of all K_{2,4}-minor-free graphs. To prove both of these results we first provide a characterization of rooted-K_{2,2}-minor-free graphs. We also prove several useful results concerning Hamilton paths in rooted K_{2,2}-minor-free graphs. |
author2 |
Xiaoya Zha |
author_facet |
Xiaoya Zha Marshall, Emily Abernethy |
author |
Marshall, Emily Abernethy |
author_sort |
Marshall, Emily Abernethy |
title |
Hamiltonicity and Structure of Classes of Minor-Free Graphs |
title_short |
Hamiltonicity and Structure of Classes of Minor-Free Graphs |
title_full |
Hamiltonicity and Structure of Classes of Minor-Free Graphs |
title_fullStr |
Hamiltonicity and Structure of Classes of Minor-Free Graphs |
title_full_unstemmed |
Hamiltonicity and Structure of Classes of Minor-Free Graphs |
title_sort |
hamiltonicity and structure of classes of minor-free graphs |
publisher |
VANDERBILT |
publishDate |
2014 |
url |
http://etd.library.vanderbilt.edu/available/etd-03212014-152116/ |
work_keys_str_mv |
AT marshallemilyabernethy hamiltonicityandstructureofclassesofminorfreegraphs |
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