Date of Award

2001

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Guoli Ding

Abstract

Graphs are characterized by whether or not they have orientations to avoid one or more of the digraphs K&ar;3 , S&ar;3 , and P&ar;3 . K&ar;3 , S&ar;3 and P&ar;3 are created by starting with a triangle, a three point star, or a path of length three respectively, and replacing each edge with a pair of arcs in opposite directions. Conditions are described when all orientations of 3-connected and 4-connected graphs must have one or more of the above digraphs as a minor. It is shown that double wheels, and double wheels without an axle, are the only 4-connected graphs with an orientation not having a K&ar;3 -minor. For S&ar;3 , it is shown that the only 4-connected graphs which may be oriented without the minor are K5 and C26 . It is also shown that all 3-connected graphs which do not have a W5-minor have an orientation without-an S&ar;3 -minor, while every orientation of a graph with a W 6-minor has an S&ar;3 -minor. It is demonstrated that K5, C26 , and C26 plus an edge are the only 4-connected graphs with an orientation without a P&ar;3 -minor. Additionally, some restrictions on large 3-connected graphs without a P&ar;3 -minor are given, and it is shown that if a 3-connected graph has a large wheel as a minor and has an orientation without a P&ar;3 -minor, then the graph must be a wheel. Certain smaller digraphs P&ar;1 , P&ar;2 , and M = K&ar;3 \a are also considered as possible minors of orientations of graphs. It is shown that a graph has an orientation without a P&ar;1 -minor if and only if it is a forest. It is shown that every orientation of a graph has a P&ar;2 -minor if and only if the graph has T2 or K+4 as a minor. To describe graphs with an orientation without an M-minor, a similar small list of graphs is given, and it is shown that if none of the given graphs is a minor of a graph, then that graph has an orientation without an M-minor.

ISBN

9780493213781

Pages

91

DOI

10.31390/gradschool_disstheses.237

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