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 paper, we give a proof of the Kelmans-Seymour conjecture. We also discuss several related results and problems.
IntroductionFor a graph G, we use T G to denote a subdivision of G, and the vertices in T G that correspond to the vertices of G are said to be its branch vertices. Thus, T K 5 denotes a subdivision of K 5 , and the vertices in a T K 5 of degree four are its branch vertices.The well known result of Kuratowski [18] states that a graph is planar if, and only if, it does not contain T K 5 or T K 3,3 . A simple application of Euler's formula for planar graphs shows that, for n ≥ 3, if an n-vertex graph has at least 3n − 5 edges then it must be nonplanar and, hence, contains T K 5 or T K 3,3 . Dirac [5] conjectured that for n ≥ 3, if an n-vertex graph has at least 3n − 5 edges then it must contain T K 5 . This conjecture was also reported by Erdős and Hajnal [7]. Kézdy and McGuiness [15] showed that a minimal counterexample to Dirac's conjecture must be 5-connected and contains K − 4 (obtained from the complete graph K 4 by deleting an edge). After some partial results in [28,30,32,33], Dirac's conjecture was proved by Mader [22], where he also showed that every 5-connected n-vertex graph with at least 3n − 6 edges contains T K 5 or K − 4 . Seymour [25] (also see [22,33]) and, independently, Kelmans [14] conjectured that every 5-connected nonplanar graph contains T K 5 . Thus, the Kelmans-Seymour conjecture implies Mader's theorem. This conjecture is also related to several interesting problems, which we will discuss in Section 7.The authors [9-11] produced lemmas needed for proving this Kelmans-Seymour conjecture, and we are now ready to prove it in this paper. Theorem 1.1 Every 5-connected non-planar graph contains T K 5 . The starting point of our work is the following result of Ma and Yu [20, 21]: Every 5-connected nonplanar graph containing K − 4 has a T K 5 . This result, combined with the result of Kézdy and McGuiness [15] on minimal counterexamples to Dirac's conjecture, gives an alternative proof of Mader's theorem. Also using this result, Aigner-Horev [1]proved that every 5-connected nonplanar apex graph contains T K 5 . A simpler proof of Aigner-Horev's result using discharging argument was obtained by Ma, Thomas and Yu, and, independently, by Kawarabayashi, see [13].We now briefly describe the process for proving Theorem 1.1. For a more detailed version, we recommend the reader to read Section 6 first, which should also give motivation to some of the technical lemmas listed in Sections 2, 3, 4 and 5.Suppose G is a 5-connected non-...
