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

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 $d\ge 425$

Long paths in random Apollonian networks

We consider the length $L(n)$ of the longest path in a randomly generated Apollonian Network (ApN) ${\cal A}_n$. We show that w.h.p. $L(n)\leq ne^{-\log^cn}$ for any constant $c<2/3$

Spanning trees without adjacent vertices of degree 2

Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist highly connected graphs in which every spanning tree contains vertices of degree 2. Using a result of Alon and Wormald, we show that there exists a natural number $d$ such that every graph of minimum degree at least $d$ contains a spanning tree without adjacent vertices of degree 2. Moreover, we prove that every graph with minimum degree at least 3 has a spanning tree without three consecutive vertices of degree 2

Counting cycles in planar triangulations

We investigate the minimum number of cycles of specified lengths in planar $n$-vertex triangulations $G$. It is proven that this number is $\Omega(n)$ for any cycle length at most $3 + \max \{ {\rm rad}(G^*), \lceil (\frac{n-3}{2})^{\log_32} \rceil \}$, where ${\rm rad}(G^*)$ denotes the radius of the triangulation's dual, which is at least logarithmic but can be linear in the order of the triangulation. We also show that there exist planar hamiltonian $n$-vertex triangulations containing $O(n)$ many $k$-cycles for any $k \in \{ \lceil n - \sqrt[5]{n} \rceil, \ldots, n \}$. Furthermore, we prove that planar 4-connected $n$-vertex triangulations contain $\Omega(n)$ many $k$-cycles for every $k \in \{ 3, \ldots, n \}$, and that, under certain additional conditions, they contain $\Omega(n^2)$ $k$-cycles for many values of $k$, including $n$

The circumference of a graph with no K3, t-minor

It was shown by Chen and Yu that every 3-connected planar graph G contains a cycle of length at least | G |log 3 2, where | G | denotes the number of vertices of G. Thomas made a conjecture in a more general setting: there exists a function β (t) > 0 for t ≥ 3, such that every 3-connected graph G with no K3, t-minor, t ≥ 3, contains a cycle of length at least | G |β (t). We prove that this conjecture is true with β (t) = log8 t t + 1 2. We also show that every 2-connected graph with no K2, t-minor, t ≥ 3, contains a cycle of length at least | G | / tt - 1. © 2006 Elsevier Inc. All rights reserved.preprin

Dense circuit graphs and the planar Tur\'an number of a cycle

The $\textit{planar Tur\'an number}$ $\textrm{ex}_{\mathcal P}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_k$ denote the cycle of length $k$. The planar Tur\'an number $\textrm{ex}_{\mathcal P}(n,C_k)$ is known for $k\le 7$. We show that dense planar graphs with a certain connectivity property (known as circuit graphs) contain large near triangulations, and we use this result to obtain consequences for planar Tur\'an numbers. In particular, we prove that there is a constant $D$ so that $\textrm{ex}_{\mathcal P}(n,C_k) \le 3n - 6 - Dn/k^{\log_2^3}$ for all $k, n\ge 4$. When $k \ge 11$ this bound is tight up to the constant $D$ and proves a conjecture of Cranston, Lidick\'y, Liu, and Shantanam

The Largest Bond in 3-Connected Graphs

A graph G is connected if given any two vertices, there is a path between them. A bond B is a minimal edge set in G such that G − B has more components than G. We say that a connected graph is dual Hamiltonian if its largest bond has size |E(G)|−|V (G)|+2. In this thesis we verify the conjecture that any simple 3-connected graph G has a largest bond with size at least Ω(nlog32) (Ding, Dziobiak, Wu, 2015 [3]) for a variety of graph classes including planar graphs, complete graphs, ladders, Mo ̈bius ladders and circular ladders, complete bipartite graphs, some unique (3,g)- cages, the generalized Petersen graph, and some small hypercubes. We will also go further to prove that a variety of these graph classes not only satisfy the conjecture, but are also dual Hamiltonian