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

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 $d\ge 425$

w-Cycles in Surface Groups

For w an element in the fundamental group of a closed, orientable, hyperbolic surface Ω which is not a proper power, and Σ a surface immersing in Ω, we show that the number of distinct lifts of w to Σ is bounded above by -χ(Σ). In special cases which can be characterised by interdependencies of the lifts of w, we find a stronger bound, whereby the total degree of covering from curves in Σ representing the lifts to the curve representing w is also bounded above by -χ(Σ). This is achieved by a method we introduce for decomposing surfaces into pieces that behave similarly to graphs, and using them to estimate Euler characteristics using a stacking argument of the kind introduced by Louder and Wilton. We then consider some consequences of these bounds for quotients of orientable surface groups by a single element. We demonstrate ways in which these groups behave analogously to one-relator groups; in particular, the ones with torsion are coherent (i.e. all finitely-generated subgroups have finite presentations), and those without torsion possess the related property of non-positive immersions as introduced by Wise

Entropy in Dimension One

This paper completely classifies which numbers arise as the topological entropy associated to postcritically finite self-maps of the unit interval. Specifically, a positive real number h is the topological entropy of a postcritically finite self-map of the unit interval if and only if exp(h) is an algebraic integer that is at least as large as the absolute value of any of the conjugates of exp(h); that is, if exp(h) is a weak Perron number. The postcritically finite map may be chosen to be a polynomial all of whose critical points are in the interval (0,1). This paper also proves that the weak Perron numbers are precisely the numbers that arise as exp(h), where h is the topological entropy associated to ergodic train track representatives of outer automorphisms of a free group.Comment: 38 pages, 15 figures. This paper was completed by the author before his death, and was uploaded by Dylan Thurston. A version including endnotes by John Milnor will appear in the proceedings of the Banff conference on Frontiers in Complex Dynamic

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application