Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees
نویسندگان
چکیده
A tree T with no vertex of degree 2 is called a homeomorphically irreducible tree (HIT) and if T is spanning in a graph, then T is called a homeomorphically irreducible spanning tree (HIST). Albertson, Berman, Hutchinson and Thomassen asked if every triangulation of at least 4 vertices has a HIST and 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 Halin graphs, which is a class of graphs closely related to HITs and HISTs. A 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 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 ≥ 4, there exists a positive real number α 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 αnd−1 . They showed that the exponent in the bound is best possible if the conjecture is true. The conjecture is confirmed for d ≥ 425. INDEXWORDS: Homeomorphically irreducible spanning tree, Halin graph, Genaralized Halin graph, 3-connected graphs, Tutte decomposition. Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees
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