113 research outputs found
Extremal Infinite Graph Theory
We survey various aspects of infinite extremal graph theory and prove several
new results. The lead role play the parameters connectivity and degree. This
includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure
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
All graphs have tree-decompositions displaying their topological ends
We show that every connected graph has a spanning tree that displays all its
topological ends. This proves a 1964 conjecture of Halin in corrected form, and
settles a problem of Diestel from 1992
A representation theorem for end spaces of infinite graphs
End spaces of infinite graphs sit at the interface between graph theory,
group theory and topology. They arise as the boundary of an infinite graph in a
standard sense generalising the theory of the Freudenthal boundary developed by
Freudenthal and Hopf in the 1940's for infinite groups.
A long-standing quest in infinite graph theory with a rich body of literature
seeks to describe the possible end structures of graphs by a set of
low-complexity representatives. In this paper we present a solution to this
fifty-year-old problem by showing that every end space is homeomorphic to the
end space of some (uniform graph on a) special order tree.Comment: 23 pages. V2 adds a moreover-part to Theorem 3.
Spanning Halin Subgraphs Involving Forbidden Subgraphs
In structural graph theory, connectivity is an important notation with a lot of applications. Tutte, in 1961, showed that a simple graph is 3-connected if and only if it can be generated from a wheel graph by repeatedly adding edges between nonadjacent vertices and applying vertex splitting. In 1971, Halin constructed a class of edge-minimal 3-connected planar graphs, which are a generalization of wheel graphs and later were named âHalin graphsâ by Lovasz and Plummer. A Halin graph is obtained from a plane embedding of a tree with no stems having degree 2 by adding a cycle through its leaves in the natural order determined according to the embedding. Since Halin graphs were introduced, many useful properties, such as Hamiltonian, hamiltonian-connected and pancyclic, have been discovered. Hence, it will reveal many properties of a graph if we know the graph contains a spanning Halin subgraph. But unfortunately, until now, there is no positive result showing under which conditions a graph contains a spanning Halin subgraph. In this thesis, we characterize all forbidden pairs implying graphs containing spanning Halin subgraphs. Consequently, we provide a complete proof conjecture of Chen et al. Our proofs are based on Chudnovsky and Seymourâs decomposition theorem of claw-free graphs, which were published recently in a series of papers
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