Georgia State University ScholarWorks @ Georgia State University Mathematics Dissertations Department of Mathematics and Statistics 5-11-2015 Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees Songling Shan Follow this and additional works at: https://scholarworks.gsu.edu/math_diss Recommended Citation Shan, Songling, "Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees." Dissertation, Georgia State University, 2015. https://scholarworks.gsu.edu/math_diss/23 This Dissertation is brought to you for free and open access by the Department of Mathematics and Statistics at ScholarWorks @ Georgia State University. It has been accepted for inclusion in Mathematics Dissertations by an authorized administrator of ScholarWorks @ Georgia State University. For more information, please contact
[email protected]. Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees by Songling Shan Under the Direction of Guantao Chen, PhD ABSTRACT 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.