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Wang, Yan

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    Subdivisions of complete graphs
    (Georgia Institute of Technology, 2017-05-23) Wang, Yan
    A subdivision of a graph G, also known as a topological G and denoted by TG, is a graph obtained from G by replacing certain edges of G with internally vertex-disjoint paths. This dissertation studies a problem in structural graph theory regarding subdivisions of a complete graph in graphs. In this dissertation, we focus on TK_5, or subdivisions of K_5. A well known theorem of Kuratowski in 1932 states that a graph is planar if, and only if, it does not contain a subdivision of K_5 or K_{3,3}. Wagner proved in 1937 that if a graph other than K_5 does not contain any subdivision of K_{3,3} then it is planar or it admits a cut of size at most 2. Kelmans and, independently, Seymour conjectured in the 1970s that if a graph does not contain any subdivision of K_5 then it is planar or it admits a cut of size at most 4. In this dissertation, we give a proof of the Kelmans-Seymour conjecture.