## LSU Historical Dissertations and Theses

#### Title

Spikes in Matroid Theory.

1998

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

James Oxley

#### Abstract

For an integer $n\ge 3,$ a rank-n matroid is called an n-spike if it consists of n three-point lines through a common point t such that, for all k in $(1, 2,\..., n - 1),$ the union of every set of k of these Lines has rank $k + 1.$ The point t is called the tip of the n-spike. Ding, Oporowski, Oxley, and Vertigan proved that, for all $n\ge 3,$ there is an integer $N(n)$ such that every 3-connected matroid with at least $N(n)$ elements has a minor isomorphic to a wheel or whirl of rank $n,\ M(K\sb{3,n})$ or its dual, $U\sb{2,n+2}$ or its dual, or a rank-n spike. In the first chapter of this dissertation, we characterise each of these classes of unavoidable matroids in terms of an extremal connectivity condition. In particular, it is proved in this chapter that if M is a 3-connected matroid of rank at least seven for which every single-element deletion or contraction is 3-connected but no 2-element deletion or contraction is, then M is a spike with its tip deleted. It is further proved that if M is a 3-connected matroid of rank at least four for which every single-element deletion is 3-connected but no 1-element contraction or 2-element deletion is, then $M\cong M\sp*(K\sb{3,n}).$. The second chapter of this dissertation evaluates the number of n-spikes representable over finite fields. It is well known that there is a unique binary n-spike for each integer $n\ge 3.$ In this chapter, we first prave that, for each integer $n\ge 3,$ there are exactly two distinct ternary n-spikes, and there are exactly $\lfloor{{n\sp2+6n+24}\over{12}}\rfloor$ quaternary n-spikes. Then we prove that, for each integer $n\ge 4,$ there are exactly $n + 2 + \lfloor {{n}\over{2}}\rfloor$ quinternary n-spikes and, for each integer $n\ge 18,$ the number of n-spikes representable over $GF(7)$ is $\lfloor{{2n\sp2+6n+6}\over{3}}\rfloor$. Finally, for each $q\ge 7,$ we find the asymptotic value of the number of distinct rank-n spikes that are representable over $GF(q)$.

9780591998146

98

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