Date of Award

5-11-2015

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Guantao Chen

Second Advisor

Ron Gould

Third Advisor

Frank Hall

Fourth Advisor

Zhongshan Li

Fifth Advisor

Hendricus Van der Holst

Sixth Advisor

Yi Zhao

Abstract

A tree $T$ with no vertex of degree 2 is called a {\it homeomorphically irreducible tree}\,(HIT) and if $T$ is spanning in a graph, then $T$ is called a {\it homeomorphically irreducible spanning tree}\,(HIST). Albertson, Berman, Hutchinson and Thomassen asked {\it if every triangulation of at least 4 vertices has a HIST} and {\it if every connected graph with each edge in at least two triangles contains a HIST}. These two questions were restated as two conjectures by Archdeacon in 2009. The first part of this dissertation gives a proof for each of the two conjectures. The second part focuses on some problems about {\it Halin graphs}, which is a class of graphs closely related to HITs and HISTs. A {\it Halin graph} is obtained from a plane embedding of a HIT of at least 4 vertices by connecting its leaves into a cycle following the cyclic order determined by the embedding. And a {\it generalized Halin graph} is obtained from a HIT of at least 4 vertices by connecting the leaves into a cycle. Let $G$ be a sufficiently large $n$-vertex graph. Applying the Regularity Lemma and the Blow-up Lemma, it is shown that $G$ contains a spanning Halin subgraph if it has minimum degree at least $(n+1)/2$ and $G$ contains a spanning generalized Halin subgraph if it is 3-connected and has minimum degree at least $(2n+3)/5$. The minimum degree conditions are best possible. The last part estimates the length of longest cycles in 3-connected graphs with bounded maximum degrees. In 1993 Jackson and Wormald conjectured that for any positive integer $d\ge 4$, there exists a positive real number $\alpha$ depending only on $d$ such that if $G$ is a 3-connected $n$-vertex graph with maximum degree $d$, then $G$ has a cycle of length at least $\alpha n^{\log_{d-1} 2}$. They showed that the exponent in the bound is best possible if the conjecture is true. The conjecture is confirmed for $d\ge 425$.

DOI

https://doi.org/10.57709/7030999

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