# Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees

#### 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 $d\ge 425$.

*This paper has been withdrawn.*