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Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees
A tree with no vertex of degree 2 is called a {\it homeomorphically irreducible tree}\,(HIT) and if is spanning in a graph, then 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 be a sufficiently large -vertex graph. Applying the Regularity Lemma and the Blow-up Lemma, it is shown that contains a spanning Halin subgraph if it has minimum degree at least and contains a spanning generalized Halin subgraph if it is 3-connected and has minimum degree at least . 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 , there exists a positive real number depending only on such that if is a 3-connected -vertex graph with maximum degree , then has a cycle of length at least . They showed that the exponent in the bound is best possible if the conjecture is true. The conjecture is confirmed for
Embedding nearly-spanning bounded degree trees
We derive a sufficient condition for a sparse graph G on n vertices to
contain a copy of a tree T of maximum degree at most d on (1-\epsilon)n
vertices, in terms of the expansion properties of G. As a result we show that
for fixed d\geq 2 and 0<\epsilon<1, there exists a constant c=c(d,\epsilon)
such that a random graph G(n,c/n) contains almost surely a copy of every tree T
on (1-\epsilon)n vertices with maximum degree at most d. We also prove that if
an (n,D,\lambda)-graph G (i.e., a D-regular graph on n vertices all of whose
eigenvalues, except the first one, are at most \lambda in their absolute
values) has large enough spectral gap D/\lambda as a function of d and
\epsilon, then G has a copy of every tree T as above
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